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## Monday, November 21, 2016

### Extremal black holes via Killing spinors

The usual way to obtain the metric and electromagnetic field of a charged black hole in general relativity is to make a spherically symmetric ansatz, insert this ansatz into the Einstein-Maxwell equations and then solve the resulting set of non-linear ordinary differential equations. In this post I explain an alternative method that uses Killing spinors. This method can be used for extremal black holes. These are black holes with charge equal to the mass.
The method is based on the following fact (see for example [1])
If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and there is a spinor $\psi$ such that $$\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0$$ and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}
A spinor $\psi$ satisfying equation \eqref{eq:20161115b} is called a Killing spinor. Definitions and conventions about the formulae above can be found at the bottom of this post.
I take the following ansatz for the metric and electromagnetic field of an extreme black hole \begin{align*} ds^2 &= - f(x)^2 dt^2 + g(x)^2 ( dx_1^2 + dx_2^2 + dx_3^2)\\ A_{\mu} dx^{\mu} &= \frac{1}{\sqrt{4 \pi}} h(x) dt, \quad\text{with}\quad F = dA \end{align*} The functions $f(x)$, $g(x)$ and $h(x)$ do not depend on $t$. Although all calculations can be easily performed on a computer after choosing explicit gamma matrices, it gives more insight to work by hand. I take the following dual tetrad: $\theta^t = f(x) dt$ and $\theta^i = g(x) dx^i$. The components of the connection are then \begin{align*} \Gamma_{tti}&= \frac{1}{f g}\partial_i f\\ \Gamma_{ijk}&= \frac{1}{g^2} \left( - \delta_{ij} \partial_k g + \delta_{ik} \partial_j g\right) \end{align*} thus \begin{equation*} \not\Gamma_t = -\frac{1}{2} \frac{1}{f g}\partial_i f\ \gamma^{ti} \\ \end{equation*} and \begin{equation*} \not\Gamma_i = \frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik} \end{equation*} also \begin{equation*} F = \frac{1}{\sqrt{4 \pi}} \partial_i h \ dx^i dt = \frac{1}{\sqrt{4 \pi}} \partial_i h \ g^{-1} f^{-1}\ \theta^i \theta^t \end{equation*} thus \begin{equation*} \not F = \frac{2}{4} \frac{1}{\sqrt{4 \pi}} \partial_i h \ g^{-1} f^{-1} \gamma^{it} \end{equation*} The $\mu = t$ component of equation \eqref{eq:20161115b} then gives \begin{equation*} \left[ -\frac{1}{2} \frac{1}{fg} \partial_i f \ \gamma^{ti} + \frac{i}{2} \partial_i h \ g^{-1} f^{-1} \gamma^{it} \gamma_t \right]\psi = 0 \end{equation*} A solution of this equation is given by $h=f$ and $\psi$ satisfying $i \gamma_t \psi = - \psi$. The $\mu = i$ component of equation \eqref{eq:20161115b} gives \begin{equation*} \left[e_i +\frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik}+ \frac{i}{2} \partial_k h \ g^{-1} f^{-1} \gamma^{kt} \gamma_i \right]\psi = 0 \end{equation*} Using $i \gamma_t \psi = - \psi$, this equation is equivalent with \begin{equation*} \left[e_i +\frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik} - \frac{1}{2} \partial_i h \ g^{-1} f^{-1} - \frac{1}{2} \partial_k h \ g^{-1} f^{-1} \gamma^{ki} \right]\psi = 0 \end{equation*} This is solved by $e_i(\psi) -\dfrac{1}{2} \partial_i h \ g^{-1} f^{-1}\psi =0$ and $\dfrac{1}{2} \dfrac{1}{g^2} \partial_k\ g + \dfrac{1}{2} \partial_k h \ g^{-1} f^{-1}=0$. Thus $g = f^{-1}$. Writing $\psi = k(x) \epsilon$, with $\epsilon$ a constant spinor, we find $k = f ^{1/2}$. Finally, the Maxwell equation $\dfrac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} F^{\mu\nu} ) =0$ gives $\partial_i ( f^{-2} \partial_i f) =0$, thus $f^{-1}$ is a harmonic function.
All in all, if $U(x_1,x_2,x_3)$ is a harmonic function, then $f = U^{-1}$, $g = U$, $h = U^{-1}$ is a solution of the Einstein-Maxwell equations with Killing spinor $\psi = U^{-1/2} \epsilon$ where $i \gamma_0 \epsilon = - \epsilon$. The special case $U(x_1,x_2,x_3) = 1+ \frac{m}{r}$ with $r^2 = x_1^2 + x_2^2 + x_3^3$ is the Reissner–Nordström solution with mass equal to the charge. The case with general $U$ is called the Majumdar-Papapetrou solution. Notice that the equations we solved above were all linear. This is much easier than solving the non-linear Einstein equations.

This method is related to supersymmetry. The equation \eqref{eq:20161115b} expresses that the supersymmetry variation of the gravitino is zero, and therefore that the solution is supersymmetric. The method can also be generalized to higher dimensions. For example, in [2], the authors classified all supersymmetric solution of (some version of) the five-dimensional Einstein-Maxwell equations. They based their classification on an equation analogous to \eqref{eq:20161115b}.

References
1. Black hole solutions in string theory, Maeda and Nozawa, 2011. hep-th/1104.1849
2. All supersymmetric solutions of minimal supergravity in five dimensions, Gauntlett, Gutowski, Hull, Pakis and Reall, 2002. hep-th/0209114

Definitions and conventions
• $\gamma^a$ are the gamma matrices that satisfy $\{ \gamma^a, \gamma^b\} = 2 \eta^{ab}$ with $\eta^{ab}=\text{diagonal}(-1,1,1,1)$
• $\gamma^{ab} = \frac{1}{2} (\gamma^a \gamma^b - \gamma^b \gamma^a)$
• The covariant derivative of a spinor is \begin{equation*} (\nabla_{\mu} \psi)^{\alpha} = \partial_{\mu} \psi^{\alpha} + \Gamma^{\alpha}_{\mu\beta} \psi^{\beta} \end{equation*} with $\Gamma^{\alpha}_{\mu\beta} = - \frac{1}{4} \Gamma_{\mu ab} (\gamma^{ab})^{\alpha}_{\ \ \beta}$.
• $\Gamma_{\mu ab}$ are the components of the spin connection where $\omega^a_{\ \ b} = \Gamma^a_{\mu b} dx^{\mu}$ and $\Gamma_{\mu ab} = \eta{bc}\Gamma^c_{\mu a}$.

• $\not F = \frac{1}{4} F_{ij} \gamma^{ij}$.
• I also write $\not\Gamma_\mu = - \frac{1}{4} \Gamma_{\mu ij} \gamma^{ij}$, thus $\nabla_{\mu}\psi = \partial_{\mu}\psi + \not\Gamma_\mu \psi$
• Indices like $\mu$, $\nu$, ... are curved indices; indices like $i,j, \ldots$ are flat indices, and indices like $\alpha, \beta, \ldots$ are spinor indices


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# Review of The Maths Behind by Colin Beveridge

Should you ask Santa for ‘Ice Col’ Beveridge’s encyclopedic tome this festive period?

What maths is behind The Maths Behind, I wonder?

You know what the biggest problem with The Maths Behind is? ‘Ice Col’ Colin Beveridge has basically covered every possible topic we can ever blog about on the Chalkdust website. This book explores the maths behind 57 everyday events: that’s more than a year of blog posts!

If this doesn’t strike you as a particularly large problem, then I reckon you will enjoy this book.

The first thing that strikes me about this encyclopedia of mathematical modelling is that it is very pretty. The publishers produce a lot of cookery books, and their designers have done a really rather good job of bringing these fifty-or-so topics to life. Most topics cover two to four pages, with large colourful graphics.

Oh cool, stopping distances! What an interesting thing to write about.

The witty text wraps around these graphics nicely, and it’s one of the standout features of this collection. Fans of silly jokes will particularly enjoy the index:

A to B, getting from… page 7
A to X, getting from… page 151.

Colin has good stock in taking mathematics and making it accessible and entertaining. Last year he published Cracking Mathematics, and before that his Maths for Dummies series has done rather well. The podcast he has co-hosted with Dave Gale since 2013, Wrong, But Useful, perhaps has provided fertile ground to collect interesting maths applications, and to find the fun in them. I’ll let you discover what he cheekily asks Siri on page 82.

## A nice selection of topics

The book (dare I say, magnum opus) is arranged into seven themes—the human world, technology, entertainment; that sort of thing—and you are actively encouraged to dip into a topic.

I don’t know who came up with the list of topics, but it’s nice to see a book like this tackle some important and current problems, as well as some classics. Some nice graphics show the dangers of gerrymandering, and a cute use of the binomial distribution tells you it’s basically never worth voting. Ah well. Top tip, though: if you drop your toast, flip it as much as possible to offset its natural tendency to fall butter-side down.

Nice, circled equations feature on most pages (but what am I summing over, hmmm?)

Many of the topics have an important equation circled at the top of the page. And these are often not easy equations! I was particularly pleased to see the Péclet number—an important ratio in fluid mechanics—appear in a discussion of penguins huddling. The Black–Scholes equation—a partial differential equation modelling the price evolution of financial derivatives—also makes an appearance (although, aha, I spot a typo!). It’s quite hard to put a second order nonlinear PDE in a book like this, but Colin does a good job of explaining their meaning. The explanation of logarithms in the discussion about the technically-not-really-Richter scale I found satisfyingly succinct and clear.

Despite already covering 57 topics, the book is littered with extra fun facts. There are nice short biographies of some of the key players in the models. And in a section on ‘pilish’—writing such as “how I wish I could recollect of circle round”, where the length of each word gives rise to an important sequence—Colin can’t help but reference the conjectured normality of pi.

## A book for… maths talks

As someone who gives pop maths talks relatively frequently, it is extremely tempting to take this book and think “here are dozens of things I could write talks about”. Is that allowed? Would you sue me, Colin?

Perhaps some of the research for book actually started with Colin’s wide knowledge of what’s happening on the pop maths scene. The section on romance reminds me of Hannah Fry’s book; the sports chapter treads some of the same ground as Rob Eastaway. Maybe I’ve forgotten how print media works but I found myself wishing I could find out more quite often, and there aren’t any obvious references.

Those $\partial$s! Sort it out, Colin

If I may add some niggles. For some odd reason(!), the publishers have opted not to use LaTeX, and the equations are a bit of a typesetting mess. It’s a shame because the graphics are otherwise lovely. (I imagine this irks Colin just as much.)

Despite the British spelling, there are -izes, Titles Capped Up Like This, and inconsistent mixes of metric and imperial units. It’s just a bit awkward to have sentences like

“An American pool ball is 5.7cm (2¼ inches) in diameter, while the mouths of the corner pockets are…between 11.4–11.8cm (4¼ and 4⅝ inches); the side pockets are…1.3cm (half inch) wider.”

Yes, yes, the pedant in me appreciates the correct use of fractional inches rather than decimal inches, but this is one of those traits of cookery books that didn’t really need to be carried over. Is anyone really using $g=32\mathrm{ft}/\mathrm{s}^2$ any more?! I suspect, given the pages on baseball and American football, that this is going pretty much unedited into the American market.

Those of us in the maths outreach community often tell people that ‘maths is in everything’. But if we’re not careful, in trying to explain how, we sweep the maths under the carpet. At the beginning of this book, Colin says that you don’t need any mathematical knowledge to read this book. For most of the book, this is probably true, and in fact the book manages to walk the line between being readable and demonstrating how it works very well. It’s not shy in showing you the actual mathematics, and it demonstrates how to present this in an accessible way.

It’s a smart way to appeal to both maths fans and the sci-curious. Pop it in a stocking, or pop it in your toilet this Christmas.

Ooh, dragon curves are really cool too.

The Maths Behind by Colin Beveridge is published by Octopus Books and retails at £14.99.

Adam is a postdoctoral researcher at Imperial College London, where he investigates weird, non-Newtonian fluids. If he’s not talking about the maths of chocolate fountains he is probably thinking about fonts, helping Professor Dirichlet answer your personal problems, and/or listening to BBC Radio 2.

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Write down a quadratic. What is the probability that it factorises? Paging Prof. Dirichet...
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We take a look at the top 10 emojis!
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Unexpected item in bagging areAAAARGGGHH here's 90p change in pennies
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Not the new coin we want, but the new coin we need

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# Necessary and sufficient condition for convergent series

Let $(a_i)_{i \in \mathbb{N}}$ be a sequence of positive reals such that $$\limsup_{i \rightarrow \infty} a_i \, i =0.$$ Is this condition necessary and sufficient for $\sum\limits_{i=1}^\infty a_i < \infty$?

Of course, if $a_i = 1/i$, then the series is infinite and if $a_i = (1/i)^{1 + \varepsilon}$, for some $\epsilon >0$, then it is finite, but it is not clear to me what happens if I choose a sequence which decays faster than $1/i$ but not faster than $(1/i)^{1 + \varepsilon}$ for arbitrary small $\varepsilon>0$.

• For the latter, consider the two series $$\sum_{n=2}^\infty\frac{1}{n\log n}\qquad\text{and}\qquad\sum_{n=2}^\infty\frac{1}{n\log^2n}.$$ The left-hand sum diverges still while the right-hand sum converges. – Clayton Aug 8 '18 at 19:25
• Incidentally $\limsup a_i/i=0$ is the same as $\lim a_i/i=0$ – zhw. Aug 8 '18 at 19:53

## 2 Answers

Note that $\sum_{n=2}^\infty\frac1{n\log n}$ diverges (this follows from the integral test), but $\lim_{n\to\infty}n\frac1{n\log n}=0$.

• Sorry, I actually meant $a_i \, i$ instead of $a_i / i$. – QuantumLogarithm Aug 13 '18 at 19:19
• @QuantumLogarithm Then you should edit your question, right?! – José Carlos Santos Aug 13 '18 at 19:21
• @QuantumLogarithm I have answered to the new version of the question. – José Carlos Santos Aug 13 '18 at 19:28

It's obviously necessary: if $\sum_na_n<\infty$, then $a_n\to0$, so $a_n/n\to0$.

It is really far from sufficient: take $a_n=\sqrt{n}$; then $a_n/n\to0$, while the series diverges spectacularly.


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## Damped Sine Wave Excel

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Therefore if we set up the waveform generator to produce a square wave and plot the signal after an edge in the waveform, we would expect to see a damped sinusoidal waveform, ie "ringing", the frequency of which can calculated using a standard formula Fig [9] Get Info; Frequency Table for Grouped Data - Online Math Learning. I used the following code: Thanks for contributing an answer to Mathematica Stack Exchange!. From the look of it, I would say cosine because it starts at a maximum point. 3 Capon Method for Exponentially Damped Sinusoidal Signals 241. Miller 2 1 Sandia National Laboratories, P. First, we need to declare some variables to store the time and the y value. I have a table of xy data which needs to be modelled using a damped sine wave. Right click on the cursor legend and choose "Copy Values" to paste the maximum value into Excel. The assumed response is in terms of a sine wave and a cosine wave. VFD produces a non pure sine wave output and the meter’s response is different for any non sine wave input. ARMA (p,q) Approaches to Economic The Box-Jenkins Approach-Diagnostic Checking Checkin Forecasting How do we know that the model we estimated is a. (a) (i) Name the effect being demonstrated. Note that in Arduino, “time” is a protected keyword, and we cannot use it to store the time. For example, when the damping factor is 0. Show the 3 values that command the sine wave on screen (amplitude, frequency, speed). A sine wave depicts a reoccurring change or motion. We'll start researching and adding these patches to our etudes. The frequency is 4kHz. I plot each one of them separately and its looks good. Fn = 1 shows the transform of damped exponent f(t) = e-at. There are two major types of electronic oscillators: harmonic oscillators and relaxation oscillators. In the "aperiodic" system as implemented in the 90P’s, damping material sits directly behind the drivers, immediately damping not only the driver movement, but the back wave that would drive the reflex system as well. Hey, Thank you for your help, I really appreciate it! Isro. In the previous section we applied separation of variables to several partial differential equations and reduced the problem down to needing to solve two ordinary differential equations. Most tissues outside the SCN also contain local clocks (the so-called peripheral clocks), and their rhythms are synchronized, harmoniously, by an array of direct or indirect signals from the SCN 2,3. It is stable. Your sine wave has a peak to peak voltage of 15-5 = 10V. I thought about just drawing them in MSVisio but then decided that it would be fun to let Excel do the work for me. If the values of the three parameters are less than 10, the. 9 MHz, (E) , Leave your comments or Download question paper. ) A damped wave is calculated using the formula: where t is time and k the damping constant. 4 Forced vibration of damped, single degree of freedom, linear spring mass systems. But this syntax doesn't seem to be directly compatible with a spreadsheet document. The former is traditionally an inductor capacitor circuit which, particularly at audio frequencies, can be cumbersome and expensive and have a response. Here is the curve y = sin x. pyplot as plt plt. At the top right of the page (or directly to the right on some browsers), you’ll see a search button. The best way to find particular calculus definitions is to perform a site search. A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions and equilibrium point to be zero. The last two cases are the ones in which we will have the most interest, as they occur repeatedly in practical. 8; % initial. To determine the temp. Since nearly all physical systems involve considerations such as air resistance, friction, and intermolecular forces where energy in the system is lost to heat or sound, accounting for damping is important in realistic oscillatory systems. 5kHz Fast PWM, 32 bit maths and a 256 sine wave lookup table (and variations in between). A calculator exists for interpolation &/or extrapolation of given data. 1 v 2 ∂ 2 y ∂ t 2 = ∂ 2 y ∂ x 2, \frac {1} {v^2} \frac {\partial^2 y} {\partial t^2} = \frac. 181 EXCEL ES 801 ARM (C) A. Addition of a Sine Wave and a Cosine Wave - GeoGebra Filter Frames of a Noisy Sine Wave Signal in Simulink - MATLAB Why AC supply is always represented by a SINE wave | you will not Sine Waves Show the Power of Perspective - WSJ How to Build a Square Wave to Sine Wave Converter Circuit | Sine. Graph of sin x. They also have an SMSC opportunity and space for student reflection. This is the currently selected item. Add an up/down control that controls the width of the stroke (of course, this would require you to discontinue the use of SetPixel()). The SAC can die down in (i) a damped exponential fashion (ii) a damped sine-wave fashion (iii) a fashion dominated by either one of or a combination of both (i) and (ii). Fn = 1 shows the transform of damped exponent f(t) = e-at. ) Except a damped sine wave decays back into a flat line. 60hz oscillator. Step by step. Homework Statement Evaluate the Fourier Transform of the damped sinusoidal wave $g(t)=e^{-t}sin(2\pi f_ct)u(t)$ where u(t) is the unit step function. For investigations a pulse generator was used generating peak currents up to 120 kA at a maximum voltage of 30 kV with a period length of 5 {micro}s of a weakly damped sine wave with 90% current reversal. k-Space Associates, Inc. Awesome Oscillator shows the difference in between the 5 SMA and 34 SMA. When a damped oscillator is subject to a damping force which is linearly dependent upon the velocity, such as viscous damping, the oscillation will have exponential decay terms which depend upon a damping coefficient. Sine waves describe many oscillating phenomena. What ever special profile you will need to test autowave. Remove the SD card and use Excel to plot the data in the file. 01 Hz to 1 MHz or more, take a look at the XR-2206. Ultra-precise on-axis encoder self-calibration for fast rotary platforms: Creator: Amin-Shahidi, Darya: Publisher: University of British Columbia: Date Issued: 2009: Description: This thesis presents the design, implementation, and experimental testing of a rotary platform and its use for encoder calibration research. Creating A Sine Wave In Excel. The double-exponential "envelope" (of the overall pulse growth and decay) is shown dashed. Free running sleep provides the best way to maximize the alertness throughout a waking day. The Sine Wave is a technical analysis tool based on advanced mathematics that indicates if a market is trending or in a cycle mode. When measuring the height of a graph, you measure the distance between the maximum crest and the minimum wave. To measure the frequency of a sine-wave voltage obtain from signal generator and to obtain lissajous pattern on the CRO screen by feeding two sine wave voltage from two signal generator. 4 06 Graphs Of Sine And Cosine. An insurance company is reviewing its current policy rates. Also called WAVE OF DISPLACEMENT. 18 bronze badges. by the charge on the capacitor over time is actually a damped sine wave. 60hz oscillator. Resembles a damped sine wave. The creation of the sine wave and both voltage measurements are synchronized such that magnitude and phase relation of the two measured voltages can be derived. Use this utility to simulate the Transfer Function for filters at a given frequency, damping ratio ζ, Q or values of R and C. The Box-Jenkins Methodology for Time Series Models Theresa Hoang Diem Ngo, Warner Bros. Antenna Wire Vibration MAE 5093 Charles O’Neill 7 June 2002 Abstract A finite difference one-dimensional wave equation solver was developed to solve forced and free vibrations of an antenna wire. MegaPulse Antenna Surge PF. It allows the frequency (1-800 Hz) and amplitude of the sine wave to be varied. A is always taken as positive, and so the amplitude of oscillation formula is just the magnitude of the displacement from the mean position. Bipolar NEFO was a damped sine wave with 140 microsecs first phase duration at 50% height; the peak amplitude of phases 2-4 decreased to 35% ,12%, and 7% of the first phase. The Chart component supports two numeric axis types: linear (or continuous) and logarithmic. Technoblogy - Tiny Function Generator Sine Wave Electrical Waveforms and Electrical Signals People The Sine Wave of History - International Man HIGH-SPEED DIGITAL DESIGN (PART 2) | SIGMADESIGN Addition of a Sine Wave and a Cosine Wave - GeoGebra. Sine waves describe many oscillating phenomena. MA (q) Significant spikes through lags q Exponential decay. Oscilloscopes for PDV: Getting More out of your Scope Stephen Mueller Applications Engineering Manager Measurements on Excel -generated trace Trace generated in Excel Acquired Trace. Often the peak of each wave decreases or dampens as time goes on. org are unblocked. Excel Homework #3. 707, so the relationship between rms values and peak values for voltage and current is: Vrms = 0. Significant spikes through lags p. A is always taken as positive, and so the amplitude of oscillation formula is just the magnitude of the displacement from the mean position. Sine Wave Generator Investigations. 9 cm2) sourced from German high-end diaphragm manufacturer Dr. The amplitude (A) of the oscillation is defined as the maximum displacement ( xmax) of the particle on either side of its mean position, i. It would be nice to see them changing as one presses the keys. You should see a damped sine wave response on the Signal Express screen. The X-axis of the sine curve represents the time. The imaginary part of the pole location is the modal frequency - the damped natural frequency (ω d) for a free decaying os-. All other bins in the lower half (s ≠ f + 1) are zero except the. m % resonance. If you have several experimental conditions, place the first into column A, the second into column B, etc. radiated power and energy for a damped sine wave. The complex number at f + 1 (== Fourier bin) has magnitude A and phase φ. d 2 x/dt 2 =-φ 2 x (2) This equation is a differential equation which says that displacement x must be a funcyion of time such that when it's second. This Thorens has two speeds, not three. Antenna Wire Vibration MAE 5093 Charles O’Neill 7 June 2002 Abstract A finite difference one-dimensional wave equation solver was developed to solve forced and free vibrations of an antenna wire. Okay, it is finally time to completely solve a partial differential equation. There is no longer any requirement to outsource to a third party to facilitate this ofen tedious task. Solv Engineering Design Solv Portfolio of Securities Solv Maximizing Income Solv Shipping Routes Solv Product Mix Solv Staff Scheduling Solver Quick Tour goal seek complex goal seek simple BudgetTab BudgetTab C_ Integer L_ q_t_ q0 R_ t_. info) The Sine Wave Inverter Directory Submitting Software makes the submission of your website to online directories and much simpler and faster process. Sine Wave RMS Value Calculation How to find Sine Wave RMS Value Average Value. 1% or peak height, etc. 5 * x_data) + np. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Describe (quantitatively and qualitatively) the motion of a real harmonic oscillator 2. Moreover, current, not energy, is the determi-. At the peak the value it is 10V DC plus 5V amplitude equals 15V. It would be nice to see them changing as one presses the keys. Now we can finally move on to model selection. ue of its real part (σ) is the rate at which a damped oscilla- tion decays. The frequency (f) of a sine or cosine function is typically expressed in terms of the number of cycles per unit time. In statistics and probability theory, the Gaussian distribution is a continuous distribution that gives a good description of data that cluster around a mean. (More specifically, the frequency in Hz is f = FR, where R is the sampling rate. The electric waves are electromagnetic disturbances spreaded where the intensity in any point of the medium is a function of the time and the intensity in any given instant is the function of the point's position. Resembles a damped sine wave. This normalizes the x-axis with respect to the sampling rate. There is often confusion on shock measurements, mainly due to the way they are presented. S y now starts out at a ﬁnite value, rather than. Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions. The sine wave would be imposed by initially "shoving" the structure with an initial velocity. Post a small Excel sheet (not a picture) showing realistic. A sine wave depicts a reoccurring change or motion. This week, we'll look at organ, lead, keys, and arpeggiator patches. This waveform was rectified to produce unipolar NEFO by cutting off phases 2 and 4. You found the oscillation frequency in the previous experiment. The only way to draw a sine wave is either with data points (but this requirement rules it out: "I want to draw sine wave with out any data points") or via Lines & Curves like above. 39 th Annual International Reliability 99 Physics Symposium, Orlando, Florida, 2001 Non-Destructive Resonant Frequency Measurement on MEMS Actuators Norman F. This is the same as multiplying by 0. , A = OQ = OR. The waveform will repeat itself every 360° or expressed mathematically every 2π radians. We have step-by-step solutions for your textbooks written by Bartleby experts!. Comment: Swing-in errors are covering a wide fre- quency bandwidth. The effect is then further damped by cabinet bracing that also acts to not only reduce cabinet coloration, but also to limit. Determine the frequency of red light which has a wavelength of 700nm. A cycle of sine wave is complete when the position. Instead we use p = Asin(ωt + φ) or more commonly p = Acos(ωt + φ) because the only difference between the two is the value of φ. Fn sets the function of the applet. Measure the period using the stopwatch or period timer. Damped Sine Wave Wikipedia. There nothing to address from a repair standpoint, and it passed the above mentioned sine wave sweep with no sonic issues (ie; buzzes, vibrations, rattles etc. Most of them involve substitutions which transform the data into a linear regression. At all times, the output is equal to the input multiplied by some constant of proportionality. Forced damped and un-damped vibrations of s. In general, messing with a sine wave is called modulation. To gain an insight to the physical background about the damped wave equation we refer to [2] where it is stated that when the neural ﬂelds are formulated to predict neural activity using brain anatomy, one is led to the damped wave equation. 18 bronze badges. A magnetic resonance imaging system (402, 500) with magnetic field gradient coils (516) and a gradient coil power supply (320, 424, 518). A sum of up to eight sinusoids with periods specified by the user, but with unknown amplitudes and phases, is fitted to the data. I'm looking for the concept beyond the results. As a result, it will have the form of an exponentially damped sinusoid. This can only be accomplished if the modulation signal is at a substantially lower frequency than the carrier wave. Both voltage and frequency reference are fed into a modulator which simulates an AC sine wave and feeds this to the motor’s stator windings. Hence, the Fourier Transform of the complex exponential given in equation [1] is the shifted impulse in the frequency domain. As a rule of thumb, an AR(p) model can adequately model a set of data if the autocorrelation function (Acf) plot looks like an infinitely damped exponential or sine wave that tails off and the partial autocorrelation function (PAcf) plot is cut off after p lags. coefficient of resistance of platinum by Callender & Griffith’s Bridge. For EAL, sine waves were modeled with 24 points per sine wave. Sine waves describe many oscillating phenomena. A hallmark of circadian clock function is the rhythmic expression of clock genes, the functional importance of which has been revealed by recent studies. Follow 309 views (last 30 days) Big AL on 16 Feb 2016. By using Microsoft Excel, we can easily create a graph sine, cosine graphs, charts and graphs tangent of the other trigonometric functions. Control Scheme for Wide-Bandgap Motor Inverters with an Observer-Based Active Damped Sine Wave Filter. Electromagnetism: Divergence, curl and gradient - Maxwell's equations - Wave equation for electromagnetic waves - Propagation in free space - Poynting vector - Rectangular and circular wave guides. Swanson 1 , and Samuel L. The controls are “Bass Level” and “EQ”. both simple waveforms such as sine, triangle and square waves as well as complex combinations of primitive functions such as an amplitude modulated waveforms or a high frequency damped sinewave Figure 2 : Power line disturbance captured on HP54601B scope. Examples of sine waves include the oscillations produced by the suspended weight on spring and the alternating current. The Apex SL is a versatile and reliable Vibration Controller serving the basic Export to Excel: Single click to export data direct to an MS Excel Spreadsheet for off-line pro- * Damped Sine and Burst Random Waveforms (option) Resonance Dwell * Locate, Lock-on and Track Resonance by Phase. component supplier. Join us now!. 00 (fully assembled & packaged) Aoc Belt drive, synchronous motor, unmeasurably small rumble, wow and flutter of better than 0. JUCE is the C++ library to develop cross. Introduction Oscillator is an electronic circuit that generates a periodic waveform on its output without an external signal source. I want to find an equation of a sine or a cosine wave. Damped Sine Wave. Analytic solutions, use of charts, electrical analog, numerical solution. Purpose To develop an Excel spreadsheet that will satisfy the characteristic equation for a fixed free beam, to plot the displacement {y} vs. How can i draw an exponential curve for a damped sine wave? I can do it manually by taking the peak points. Synthesine is a web playground for synthesizing sound waves from scratch. 5 ))) + aes (x) + stat_function ( fun = f) + ggtitle ( "Dampened Sine Wave" ). Sine, damped Sine, Lorentz, Modified Lorentz, Power (ie Polynomial) and Exponential series are presently available models to match your data. The Family of Direct Radiators Ahem. In other words, the wave gets flatter as the x-values get larger. In the "aperiodic" system as implemented in the 90P’s, damping material sits directly behind the drivers, immediately damping not only the driver movement, but the back wave that would drive the reflex system as well. 0 0 C t Ce at C>0 and a>0. Atrial fibrillation as a self sustaining arrhythmia independent of focal discharge. The waveform has a peak-to-peak voltage of 4V and DC bias of 103uV (calculated from the signal itself). cpp: Cubic Spline Curve-fit: spline. Oscillations. damped sine wave while the PACF=0 6 Figure 1. linspace(-5, 5, num=50) y_data = 2. Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. 3 Program 6. Basic sine and cosine curves y = sin x y = cos x. The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function ( )Hj. Write the value down since you will be using it again. 59 MHz, (D) 15. MegaPulse Antenna Surge PF. We strongly suggest trying a Lorentz series for data with multiple peaks or valleys. The output of a zero order system is proportional to the input. 1 dB ≤1 MHz ±0. There are financial time series which look like damped oscillations composed with S curves (e. ‘plot_number’ can range from 1 to a maximum of ‘nrows’ * ‘ncols’. Kurt Muller (DKM). If the Value follows the curve of a damped sine wave, it will look most realistic: Damped Sine Wave. =20 It includes an Excel workbook for computing the analog equivalent=20 component values from the geophone spec. Fn = 1 shows the transform of damped exponent f(t) = e-at. Trivial once you know how to do it. Sine Wave Background supports png. Neuber’s constant for the notch sensitivity of steel 189 Fitting data to an exponentially decaying sine wave 190 First zero crossing of an exponentially decaying sine wave 192 Area of an exponentially decaying sine wave 193 Length of a line in space 194 Fourier transform of a sine wave 198 Cross correlation of two pulses 200 Lowest five. (Turn in at next weeks lab. A sine wave is characterised by three quantities 1) Amplitude A 2) Angular frequency ω (radians/s) ω = 2πf where f is in cycles/s or Hz 3) Phase φ (radians) How many radians in 360o? Period T = 1/f = 2π/ω = time for one full cycle of the motion. FREQUENCY DOMAIN AND FOURIER TRANSFORMS So, x(t) being a sinusoid means that the air pressure on our ears varies pe- riodically about some ambient pressure in a manner indicated by the sinusoid. Synthesine is a web playground for synthesizing sound waves from scratch. The reason was not only their intrinsic importance, but also that any motion can be expressed in terms of a sum of sinusoidal oscillations, using the Fourier components. boxing the compass. Create AccountorSign In. ok looks pretty simple, doesn’t it. cpp: Integrate a time history via the trapezoidal rule. (c) Using other frequencies Repeat part (a) for ~10 Hz, ~20 Hz, and ~50 Hz. List of Contents1 RLC Resonant frequency Formula1. To gain an insight to the physical background about the damped wave equation we refer to [2] where it is stated that when the neural ﬂelds are formulated to predict neural activity using brain anatomy, one is led to the damped wave equation. Strength of materials Strength of materials is the science which deals with the relations between externally applied loads and their internal effects on bodies. , A = OQ = OR. Note that in Arduino, “time” is a protected keyword, and we cannot use it to store the time. =20 It includes an Excel workbook for computing the analog equivalent=20 component values from the geophone spec. Our aim is to explore the slope of the curve y = sin x. The next component is a 1 Hz sine wave with an amplitude of 0. Freesound General-Purpose Audio Tagging Challenge. Microwaves : Properties and applications - Generation by magnetron and reflex klystron oscillator - Travelling wave tube - Biological effects. Voltmeter Frequency Response: Flat within 3 % over range of 10 to 100,000 cps. Purpose To develop an Excel spreadsheet that will satisfy the characteristic equation for a fixed free beam, to plot the displacement {y} vs. But I have yet to figure out how to do a sinusoidal regression. This banner text can have markup. These types of oscillators are used in RF generators, high-frequency heating, radio and TV receivers, etc. If >1, system is named as Over Damped System; Response of a Second order system. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. One appears to have an equation of a form: ( b - ax ) sin (ωx) rather than a damped sine wave. Here is the third part of a tutorial in both elementary dynamics and numerical methods. Either of these patterns in a PACF suggests an MA model. 0-7803-6587-9/01/$10. a parallel LC circuit which produces a damped sine wave. Lihat profil lengkap di LinkedIn. (damped sine wave) may appear if some of the 0 js are complex numbers Theorem. The mathematics of PDEs and the wave equation and we obtain the wave equation for an inhomogeneous medium, ρ·u tt = k ·u xx +k x ·u x. Bipolar NEFO was a damped sine wave with 140 microsecs first phase duration at 50% height; the peak amplitude of phases 2-4 decreased to 35% ,12%, and 7% of the first phase. So it is a sine wave with an amplitude of 5V and it rides on top of a DC level of 10V. For some MA models, the PACF shows exponential decay instead. Muhammad Nur Ariff menyenaraikan 3 pekerjaan pada profil mereka. The waveform will repeat itself every 360° or expressed mathematically every 2π radians. 707 Vo and Irms = 0. f systems vibration due to movement of supporting structures, transmissibility. It's just a sine wave. Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions. 3 Program 6. This normalizes the x-axis with respect to the sampling rate. Procedure We assume the material of the cantilever beam as steel having the following properties. The sine wave would be imposed by initially "shoving" the structure with an initial velocity. I would like to see equations, methods, so on. d 2 x/dt 2 =-φ 2 x (2) This equation is a differential equation which says that displacement x must be a funcyion of time such that when it's second. Creating A Sine Wave In Excel. 4: Your graph should show about ﬁve or six cycles of a sine wave with an amplitude of about 12Hz, hence extending from 380 to 404Hz. Long Short-Term Memory Networks With Python. Sine waves describe many oscillating phenomena. normal(size=50) # And plot it import matplotlib. David Corrigan Electronic and Electrical Engineering [email protected] a b + x sin 2 π c x + d. the slope of the curve changes as we move. Designing of an IIR Filter using MATLAB & DSP Kit 4. This technique is called pulse width modulation (PWM) and utilises the fact that there is a diode rectifier towards the mains and the intermediate DC voltage is kept constant. Graphs Of The Sine And Cosine Function Precalculus Ii. It follows that the solutions of this equation are superposable, so that if and are two solutions corresponding to different initial conditions then is a third solution, where and are arbitrary. CONFERENCES Presented the paper titled as “ Fractional-order scheme for HIV Internal Viral Dynamics ” in 1st LGU National Conference on Pure & Applied Mathematics, 17-18 May, 2017. 2 It would have been much easier to enter these data into Excel save as a tab from MATH 107 at Kwame Nkrumah Uni. Long Short-Term Memory Networks With Python. But there is another mode in which you can use an oscilloscope. Amplitude the AC value of the sine wave; Carrier Freq is the carrier frequency. It is defined by the following equations: We have to use two state variables ( x1 representing the position and x2 representing the velocity) to reduce the order of that differential equation and arrive at the following representation:. So he's given an equation of the graph that I have here. Moreover, current, not energy, is the determi-. The trend line feature of Excel is used, and exponential is selected. NumPy has the sin() function,. It is also imaginable that this is a more realistic model of an earthquake (possibly!). There are many other wave forms and shapes possible but these are the main ones encountered in Rife type work. The Beginner's Guide to Excel - Excel Basics Tutorial - Duration: 21:47. Bipolar NEFO was a damped sine wave with 140 microsecs first phase duration at 50% height; the peak amplitude of phases 2-4 decreased to 35% ,12%, and 7% of the first phase. Figures 4 and 5 show the effect of the controls on the response. pyplot as plt plt. (damped sine wave) may appear if some of the 0 js are complex numbers Theorem. The device operates in a wide frequency range from 1 Hz to 50 MHz. Echo Sequence Prediction Problem. This is the currently selected item. These types of oscillators are used in RF generators, high-frequency heating, radio and TV receivers, etc. Check that each graph shows the. When the wave is damped, each successive peak decreases. Search for jobs related to Com generator wave java or hire on the world's largest freelancing marketplace with 15m+ jobs. So, I wrote a Visual Basic routine in Excel that first generated a sampled-data sinusoid, then calculated the FFT of that waveform and plots both the waveform and the FFT. US8323219B2 US12/932,468 US93246811A US8323219B2 US 8323219 B2 US8323219 B2 US 8323219B2 US 93246811 A US93246811 A US 93246811A US 8323219 B2 US8323219 B2 US 8323219B2 Authority. By looking at three dB down from the peak level, one can determine the associated damping as shown in Figure 2. If necessary press the run/stop button and use the horizontal shift knob to get the full damped curve in view. At the bottom it is 10V DC minus 5V amplitude equals 5V. It would seem a natural sine wave - a cycle that switches from cold-to-warm-back-to-cold-etc. Fn = 1 shows the transform of damped exponent f(t) = e-at. It looks something like this. frame ( x = c ( - 3. There are financial time series which look like damped oscillations composed with S curves (e. This mathematical relationship between the average values applies to both AC. A sine wave is a repetitive change or motion which, when plotted as a graph, has the same shape as the sine function. 14 shows acceleration data with white noise. Fall 2004 3. Fn = 2 to 6 show special cases of Fn = 1. Your sine wave has a peak to peak voltage of 15-5 = 10V. answered Sep 23 '14 at 14:48. (damped sine wave) may appear if some of the 0 js are complex numbers Theorem. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A. So the data comes from blood glucose after a meal overtime, it usually starts at some value, say 90mg/dl at mealtime, peaks to 150mg/dl, then decreases down to some value which can be less than the intial 90mg/dl. The current amplitude is then measured as a function of frequency. 3: MATLAB program to find the response. Fn sets the function of the applet. 1] George Holzwarth. Multiplying a sine or cosine function by a constant changes the graph of the parent function; specifically, you change the amplitude of the graph. Google Classroom Facebook Twitter. Both of these values need to be decimal numbers. This leading indicator is also called the MESA indicator and was developed by John Ehlers based on an. Solv Portfolio of Securities Solv Maximizing Income Solv Shipping Routes Solv Product Mix. Most of them involve substitutions which transform the data into a linear regression. the physical size of a motor is. Fn = 2 to 6 show special cases of Fn = 1. Shock Response Spectrum Analysis A Shock Response Spectrum (SRS) is a graphical presentation of a transient acceleration pulse’s potential to damage a structure. Electromagnetism: Divergence, curl and gradient - Maxwell's equations - Wave equation for electromagnetic waves - Propagation in free space - Poynting vector - Rectangular and circular wave guides. After we see that the sound is correctly recognized as the sine wave at 1000 Hz, we can click the Calibrate button to perform a calibration. Add to Cart. It is easy to see that the cosine wave would be generated by imposing an initial displacement on the structure and then releasing. The resulting graph should show 5 cycles of the signal. The Beginner's Guide to Excel - Excel Basics Tutorial - Duration: 21:47. (More specifically, the frequency in Hz is f = FR, where R is the sampling rate. It's free to sign up and bid on jobs. Regarding your question, yes, it does apply. 1] George Holzwarth. 0 0 C t Ce at C>0 and a>0. View and Download LeCroy 6000 SERIES operator's manual online. white noise 5. Oscillating functions aren't. Figure 1: The notch and delay filters modulate the Earth’s albedo. Regular or linear frequency ( f ) , sometimes also denoted by the Greek symbol "nu" ( ν ), counts the number of complete oscillations or rotations in a given period of time. The left and right bounds depend on how you define them: e. 2 Example Numerical. Section 9-5 : Solving the Heat Equation. Circulation 2000; 101:1282-1287. The wavelength of the sine wave is denoted by λ. menus (Figure 2). However the motor is an AC synchronous type, electronically con-trolled. Given a general di erence equation of the form C(B)Zt = 0 where C(B) = 1+C1B +C2B2 +:::CnBn and C(B) = Qn i=1(1 RiB) so the R0is are the reciprocal roots of the equation C(B) = 0, we have that the solution is Zt = Pn i=1 AiR t i (without proof). Motion Analysis Matlab. But I have yet to figure out how to do a sinusoidal regression. a b + x sin 2 π c x + d. Forum for supporting LAVA code published on the LabVIEW Tools Network. by the charge on the capacitor over time is actually a damped sine wave. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Spring-Mass System Consider a mass attached to a wall by means of a spring. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Commented: jay shah on 8 Aug 2017. linspace(-5, 5, num=50) y_data = 2. to program you profile and adjust each single parameter of it. ) Except a damped sine wave decays back into a flat line. Moe GK, Abildskov JA. It is one of the oldest and the foremost structural engineering Journal of its kind and. The sine wave looks great but slowly oscillates. The data list includes. Note - the coils of the spring should never completely "collapse" and touch each other at any point in the oscillation. The magnetic resonance imaging system further comprises a processor (540) and a chiller (308,. Because tuning my VCO wasn't going well, I decided to adjust the numbers generated from the Excel spreadsheet so I'd have an in-tune playable synth. curve needed for the damped sine wave. Liberty Rd. Damped sine wave, a sinusoidal function whose amplitude decays as time increases. How can i draw an exponential curve for a damped sine wave? I can do it manually by taking the peak points. I've got an example. A high performance sine wave Inverter Controller with Capacitor Current Feedback and "Back EMF" Decoupling: 26th Annual IEEE Power Electronics Specialists Conference, 1995. The Wolfram Language also supports unique symbolic interpolating functions that can immediately be used throughout the system to efficiently represent approximate numerical functions. 2 Example Numerical. Video created by Berklee College of Music for the course "Creating Sounds for Electronic Music". When x = 1, this series is called the harmonic series, which increases without bound—i. Firstly want to say a few words about JUCE. to program you profile and adjust each single parameter of it. difference transforms to a sine wave with a 1/2 sample delay. Optimal differentiators can be looked at as a variant of windowing a ramp, by using linear combinations of the first M finite differences to make an M+1 order approximation to the derivative over some frequency band, but they always roll off at high frequency, which is a key point. Most regressions are easy. This signal is greatest on the phase the dimmer is on as would be expected. As the energy is radiated the string vibration is damped. The last two cases are the ones in which we will have the most interest, as they occur repeatedly in practical. Determine the wavelength λ (in terms of the number of unit cells) from the standing wave pattern and compare to the ideal, theoretical value (assuming lossless components with nominal component values). However the motor is an AC synchronous type, electronically con-trolled. It is defined by the following equations: We have to use two state variables ( x1 representing the position and x2 representing the velocity) to reduce the order of that differential equation and arrive at the following representation:. Smith 1 , Danelle M. sheet, and includes some examples. The mathematics of PDEs and the wave equation and we obtain the wave equation for an inhomogeneous medium, ρ·u tt = k ·u xx +k x ·u x. The left and right bounds depend on how you define them: e. There are many other wave forms and shapes possible but these are the main ones encountered in Rife type work. he meter is calibrated for sine wave AC. This banner text can have markup. PID Control Based on a survey of over eleven thousand controllers in the reﬁning, chemi-cals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback. 0 0 C t Ce at C>0 and a<0. The Chart component supports two numeric axis types: linear (or continuous) and logarithmic. 2; % initial position v0 = 0. S Of Trig Graphs. Amplitude (a) of a wave is how big the wave is or the energy of the wave. MATLAB Measurements 6/6/2016 17 Custom MATLAB parameter finds the half -life of the damped sine wave The value 3. Using Excel to fit a damped sinusoid to experimental data: y = A exp(-gt) cos(wt + f) [Eq. G-protein-coupled receptors (GPCRs) constitute the largest family of cell surface receptors, participating in a broad range of physiological functions. 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Pythagorean Identity Formula and Graph

Formula , Graph, Unit Circle

The picture shows a point on the unit circle. Since we know that any point on the unit circle can be described by $$cosθ$$ , $$sinθ$$. It is possible to draw the triangle that describes this formula. As you can see from the picture below, the length of one side is cosθ and the length of the other side is sinθ and, by definition, the radius of the unit circle is 1. From these facts, the primary Pythagorean identity can be shown. sin2θ + cos2θ = 1. This identity is just an application of the Pythagorean theorem to the unit circle.

Formula for the Pythagorean Identities

• $$sin^2 \theta  ; + cos^2 \theta = 1$$
• $$tan^2 \theta + 1 = sec^2 \theta$$
• $$1 + cot ^2 \theta = csc^2 \theta$$

Pictured below is the formula for the aptly named Pythagorean identity:

$$sin^2 \theta + cos^2 \theta = 1$$

Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!


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# Precession of angular momentum in a magnetic field

Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 14, Exercise 14.4.1.

[If some equations are too small to read easily, use your browser’s magnifying option (Ctrl + on Chrome, probably something similar on other browsers).]

In classical electrodynamics, the torque on a magnetic moment ${\boldsymbol{\mu}}$ in a constant magnetic field ${\mathbf{B}}$ is given by (using Shankar’s notation):

$\displaystyle \mathbf{T}=\boldsymbol{\mu}\times\mathbf{B} \ \ \ \ \ (1)$

We can relate the magnetic moment to the angular momentum of a (classically) spinning charged object by introducing the gyromagnetic ratio

$\displaystyle \gamma\equiv\frac{\mu}{l} \ \ \ \ \ (2)$

If we apply this to a single particle of charge ${q}$ and mass ${m}$ travelling at constant speed ${v}$ around a circular orbit, then its angular momentum is

$\displaystyle l=mvr \ \ \ \ \ (3)$

Tha magnetic moment can be calculated by taking the charge ${q}$ to be smeared out over the circumference of the circle, giving a linear charge density of

$\displaystyle \lambda=\frac{q}{2\pi r} \ \ \ \ \ (4)$

Since the loop is spinning with speed ${v}$, the current (rate at which charge passed a fixed point on the circle) is

$\displaystyle I=\lambda v=\frac{q}{2\pi r}r\omega=\frac{q}{2\pi}\omega \ \ \ \ \ (5)$

where

$\displaystyle \omega=\frac{2\pi}{P}=\frac{2\pi v}{2\pi r}=\frac{v}{r} \ \ \ \ \ (6)$

is the angular frequency (${P}$ is the period, or time it takes for one complete orbit).

The magnetic moment is defined as

$\displaystyle \boldsymbol{\mu}\equiv\frac{I}{c}\mathbf{a} \ \ \ \ \ (7)$

where ${\mathbf{a}}$ is the area of the loop, whose direction is determined by using the right-hand rule on the direction of the current around the loop. Thus if the current is travelling counterclockwise when viewed from above, ${\mathbf{a}}$ points upwards. (The speed of light ${c}$ enters because Shankar is using CGS units.) The magnetic moment here is then

 $\displaystyle \boldsymbol{\mu}$ $\displaystyle =$ $\displaystyle \frac{qv}{2\pi r}\frac{\pi r^{2}}{c}\hat{\mathbf{a}}\ \ \ \ \ (8)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\frac{q}{2mc}\right)\left(mvr\hat{\mathbf{a}}\right)\ \ \ \ \ (9)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\frac{q}{2mc}\right)\mathbf{l} \ \ \ \ \ (10)$

where ${\mathbf{l}}$ is the angular momentum vector. In this case, the gyromagnetic ratio is

$\displaystyle \gamma=\frac{q}{2mc} \ \ \ \ \ (11)$

In this case, the torque 1 is given by

$\displaystyle \mathbf{T}=\gamma\mathbf{l}\times\mathbf{B} \ \ \ \ \ (12)$

The interaction energy (between the angular momentum and magnetic field) is given by

$\displaystyle H_{int}=\int T\left(\theta\right)d\theta \ \ \ \ \ (13)$

where the torque is given as a function of the angle between ${\boldsymbol{\mu}}$ and ${\mathbf{B}}$ in 1, so that

$\displaystyle T=\mu B\sin\theta \ \ \ \ \ (14)$

Doing the integral (neglecting the constant of integration) we have

$\displaystyle H_{int}=-\mu B\cos\theta=-\boldsymbol{\mu}\cdot\mathbf{B} \ \ \ \ \ (15)$

${H_{int}}$ is minimized when ${\boldsymbol{\mu}}$ and ${\mathbf{B}}$ are parallel, so the torque’s effect is to try to bring these two vectors into alignment. This assumes that the magnetic moment doesn’t actually involve any angular momentum, which obviously isn’t the case with our rotating loop example above. In that case, the torque causes a precession about the direction of ${\mathbf{B}}$, which we can see as follows.

The angular version of Newton’s law, relating torque and angular momentum, is

$\displaystyle \mathbf{T}=\frac{d\mathbf{l}}{dt}=\gamma\mathbf{l}\times\mathbf{B} \ \ \ \ \ (16)$

Since the cross product is perpendicular to both its constituent vectors, the change in ${\mathbf{l}}$ is always perpendicular to ${\mathbf{l}}$ itself. The effect can be seen by looking at Shankar’s Figure 14.2 (too much effort to reproduce that here), in which we can see that

 $\displaystyle \Delta\mathbf{l}$ $\displaystyle =$ $\displaystyle \gamma\left(\mathbf{l}\times\mathbf{B}\right)\Delta t\ \ \ \ \ (17)$ $\displaystyle \Delta l$ $\displaystyle =$ $\displaystyle \gamma lB\sin\theta\Delta t \ \ \ \ \ (18)$

where ${\theta}$ is the angle between ${\mathbf{l}}$ and ${\mathbf{B}}$, and ${\Delta\mathbf{l}}$ is tangent to the circle of radius ${l\sin\theta}$ that lies in the plane perpendicular to ${\mathbf{B}}$. The net effect is that ${\mathbf{l}}$ precesses about the direction of ${\mathbf{B}}$, so that the magnitude of angular momentum remains constant, but its direction changes at a constant rate. The change in azimuthal angle ${\Delta\phi}$ in time ${\Delta t}$ is

$\displaystyle \Delta\phi=\frac{-\Delta l}{l\sin\theta}=-\gamma B\Delta t \ \ \ \ \ (19)$

where the minus sign is because the angular momentum precesses clockwise (as seen from above) around ${\mathbf{B}}$. The angular frequency of precession is therefore

$\displaystyle \omega_{0}=\frac{\Delta\phi}{\Delta t}=-\gamma B \ \ \ \ \ (20)$

If we include the direction of the axis of precession, which is parallel to ${\mathbf{B}}$, then

$\displaystyle \boldsymbol{\omega}_{0}=-\gamma\mathbf{B} \ \ \ \ \ (21)$

We can see that these results transfer over to quantum mechanics if we use Ehrenfest’s theorem. For an interaction hamiltonian 15, we can write it as

$\displaystyle H=-\gamma\mathbf{L}\cdot\mathbf{B} \ \ \ \ \ (22)$

We want to find the average of the angular momentum over time, so we use Ehrenfest’s theorem to write

$\displaystyle \frac{d\left\langle \mathbf{L}\right\rangle }{dt}=-\frac{i}{\hbar}\left\langle \left[\mathbf{L},H\right]\right\rangle \ \ \ \ \ (23)$

We can work out the RHS using the commutators of angular momentum:

$\displaystyle \left[L_{i},L_{j}\right]=i\hbar\sum_{k}\varepsilon_{ijk}L_{k} \ \ \ \ \ (24)$

As we’re dealing with a vector operator, we can work out each component separately. For ${L_{x}}$ we get, assuming that ${\mathbf{B}}$ is independent of position (and thus commutes with ${\mathbf{L}}$):

 $\displaystyle -\frac{i}{\hbar}\left[L_{x},H\right]$ $\displaystyle =$ $\displaystyle \frac{i\gamma}{\hbar}\left[L_{x},L_{x}B_{x}+L_{y}B_{y}+L_{z}B_{z}\right]\ \ \ \ \ (25)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{i\gamma}{\hbar}\left(\left[L_{x},L_{x}\right]B_{x}+\left[L_{x},L_{y}\right]B_{y}+\left[L_{x},L_{z}\right]B_{z}\right)\ \ \ \ \ (26)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -\gamma\left(0+L_{z}B_{y}-L_{y}B_{z}\right)\ \ \ \ \ (27)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \gamma\left(\mathbf{L}\times\mathbf{B}\right)_{x}\ \ \ \ \ (28)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\boldsymbol{\mu}\times\mathbf{B}\right)_{x} \ \ \ \ \ (29)$

The other two components work out similarly, so we have

$\displaystyle -\frac{i}{\hbar}\left\langle \left[\mathbf{L},H\right]\right\rangle =\boldsymbol{\mu}\times\mathbf{B} \ \ \ \ \ (30)$

As ${\mathbf{B}}$ doesn’t depend on position, when we take the average over space we get

$\displaystyle \frac{d\left\langle \mathbf{L}\right\rangle }{dt}=\left\langle \boldsymbol{\mu}\right\rangle \times\mathbf{B} \ \ \ \ \ (31)$

Thus the mean of the quantum angular momentum also precessed about ${\mathbf{B}}$. Since the only assumption we made was that ${\mathbf{B}}$ was independent of position, and all that was used in the derivation was the commutation relations of angular momentum, the result is also valid for spin angular momentum, and time-varying magnetic fields, provided they are constant over all space.


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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 8.4.4 Idempotent Completeness

We now study $\infty$-categories in which every idempotent splits.

Definition 8.4.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. We say that $\operatorname{\mathcal{C}}$ is idempotent complete if every idempotent $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ splits (see Definition 8.4.3.5).

Example 8.4.4.2. Let $\operatorname{\mathcal{C}}$ be a category. If $\operatorname{\mathcal{C}}$ admits equalizers (or coequalizers), then the $\infty$-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is idempotent complete (this is a restatement of Corollary 8.4.2.6). In particular, if $\operatorname{\mathcal{C}}$ admits finite limits or finite colimits, then $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is idempotent complete.

Warning 8.4.4.3. An $\infty$-category which admits finite limits (or colimits) need not be idempotent complete. See Example .

Example 8.4.4.4. Let $X$ be a Kan complex. Since the simplicial set $\operatorname{N}_{\bullet }(\operatorname{Idem})$ is weakly contractible (Remark 8.4.3.4), every morphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow X$ is homotopic to a constant map. It follows that $X$ is idempotent complete when viewed as an $\infty$-category.

Remark 8.4.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is idempotent complete.

Proposition 8.4.4.6. Let $\operatorname{\mathcal{C}}$ be an idempotent complete $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every object $X \in \operatorname{\mathcal{C}}_0$ and every object $Y \in \operatorname{\mathcal{C}}$ which is a retract of $X$, there exists an object $Y' \in \operatorname{\mathcal{C}}_0$ which is isomorphic to $Y$. Then $\operatorname{\mathcal{C}}_0$ is idempotent complete.

Proof. Let $\operatorname{Ret}$ denote the category of Construction 8.4.0.1. Suppose we are given an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_0$, carrying the object $\widetilde{X} \in \operatorname{Idem}$ to an object $X = F( \widetilde{X} ) \in \operatorname{\mathcal{C}}_0$. We wish to show that $F$ is a split idempotent in $\operatorname{\mathcal{C}}_0$. Since $\operatorname{\mathcal{C}}$ is idempotent complete, we can extend $F$ to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, carrying the object $\widetilde{Y} \in \operatorname{Ret}$ to an object $Y = \overline{F}( \widetilde{Y} )$ which is a retract of $X$. By assumption, we can choose an isomorphism $\alpha _0: Y \rightarrow Y'$, where $Y'$ belongs to $\operatorname{\mathcal{C}}_0$. Using Corollary 4.4.5.3, we can lift $\alpha _0$ to an isomorphism of functors $\alpha : \overline{F} \rightarrow \overline{F}'$ in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}})$, whose image in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ is the identity transformation from $F$ to itself. Then $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_0$ is a splitting of the idempotent $F$. $\square$

Proposition 8.4.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. The following conditions are equivalent:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is idempotent complete.

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits limits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.

$(3)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits colimits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.

Proof. This is an immediate consequence of Corollary 8.4.3.11. $\square$

Remark 8.4.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if the restriction functor

$\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$

is an equivalence of $\infty$-categories. This is an immediate consequence of Corollary 8.4.3.10.

Corollary 8.4.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories which are equivalent. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if $\operatorname{\mathcal{D}}$ is idempotent complete.

Corollary 8.4.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $K$ be a simplicial set. If $\operatorname{\mathcal{C}}$ is idempotent complete, then $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is idempotent complete.

Corollary 8.4.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $\operatorname{\mathcal{C}}$ is idempotent complete, then the slice and coslice $\infty$-categories $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are idempotent complete.

To apply the criterion of Proposition 8.4.4.7, it is often useful to replace $\operatorname{N}_{\bullet }( \operatorname{Idem})$ by a simpler simplicial set.

Notation 8.4.4.12. Let $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ denote the $1$-dimensional simplicial set associated to the directed graph

$\cdots \rightarrow -2 \rightarrow -1 \rightarrow 0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots$

We let $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ be the morphism of simplicial sets corresponding to the diagram

$\cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots$

in the category $\operatorname{Idem}$.

Remark 8.4.4.13. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is $1$-dimensional, the morphism $Q$ of Notation 8.4.4.12 factors (uniquely) through the $1$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$, which we can identify with the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$. Under this identification, $Q$ corresponds to a morphism of simplicial sets $q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$. This is a covering map (see Definition 3.1.4.1), which exhibits the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$ as the quotient of $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ by a free action of the group $(\operatorname{\mathbf{Z}},+)$ by translations. The induced map of geometric realizations $| \operatorname{Spine}[\operatorname{\mathbf{Z}}] | \rightarrow | \Delta ^1 / \operatorname{\partial \Delta }^1 |$ can be identified with the standard covering map $\mathbf{R} \rightarrow S^1$ in the category of topological spaces.

Remark 8.4.4.14. In the situation of Notation 8.4.4.12, we can regard $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ as a simplicial subset of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$, where we regard the set of integers $\operatorname{\mathbf{Z}}= \{ \cdots < -2 < -1 < 0 < 1 < 2 < \cdots \}$ as equipped with its usual linear ordering. Moreover, the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})$ is inner anodyne (this is a special case of Proposition 1.4.7.3).

Remark 8.4.4.15. The simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible. This follows from Remark 8.4.4.14, since the $\infty$-category $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})$ is filtered and therefore weakly contractible (Proposition 7.2.4.9). Alternatively, it can be deduced from Example 3.5.4.4, since the geometric realization $| \operatorname{Spine}[\operatorname{\mathbf{Z}}] |$ is homeomorphic to the set of real numbers $\mathbf{R}$ (endowed with its usual topology).

Proposition 8.4.4.16. The morphism $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ of Notation 8.4.4.12 is both left and right cofinal.

Proof. We will show that $Q$ is left cofinal; a similar argument will show that it is right cofinal. By virtue of Theorem 7.2.3.1, it will suffice to show that the simplicial set $K = \operatorname{Spine}[\operatorname{\mathbf{Z}}] \times _{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } \operatorname{N}_{\bullet }( \operatorname{Idem})_{/ \widetilde{X}}$ is weakly contractible. Let us identify the vertices of $K$ with pairs $(n,f)$, where $n$ is an integer and $f: \widetilde{X} \rightarrow \widetilde{X}$ is a morphism in the category $\operatorname{Idem}$. Unwinding the definitions, we see that $K$ is the $1$-dimensional simplicial set associated to the direct graph given in the diagram

$\xymatrix@R =50pt@C=50pt{ \cdots \ar [dr] \ar [r] & (-1, \widetilde{e}) \ar [dr] \ar [r] & (0, \widetilde{e}) \ar [dr] \ar [r] & (1, \widetilde{e}) \ar [dr] \ar [r] & \cdots \\ \cdots & (-1,\operatorname{id}_{\widetilde{X}} ) & (0, \operatorname{id}_{\widetilde{X}} ) & (1, \operatorname{id}_{\widetilde{X}} ) & \cdots . }$

The inclusion of the upper part of the diagram determines a monomorphism of simplicial sets $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow K$ which is left anodyne (since it is a pushout of a coproduct of countably many copies of the inclusion map $\{ 0\} \hookrightarrow \Delta ^1$), and therefore a weak homotopy equivalence (Proposition 3.1.6.14). The desired result now follows from the weak contractibility of the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ (Remark 8.4.4.15). $\square$

Corollary 8.4.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits sequential limits (or colimits). Then $\operatorname{\mathcal{C}}$ is idempotent complete.

Proof. It follows from Proposition 8.4.4.16 (and Corollary 7.2.2.12) that the $\infty$-category $\operatorname{\mathcal{C}}$ admits limits (or colimits) indexed by the $\infty$-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$, and is therefore idempotent complete by virtue of Proposition 8.4.4.7. $\square$

Remark 8.4.4.18. Broadly speaking, Proposition 8.4.4.16 will be useful to us because it shows that the $\infty$-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$ admits a (left and right) cofinal diagram $Q: K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where the simplicial set $K$ is finite-dimensional. Beware that it is not possible to arrange that the simplicial set $K$ is finite, since an $\infty$-category which admits finite colimits need not be idempotent complete (Warning 8.4.4.3). In particular, there does not exist a categorical equivalence $K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where $K$ is a finite simplicial set.

Example 8.4.4.19. Let $\operatorname{\mathcal{QC}}$ denote the $\infty$-category of (small) $\infty$-categories. Then $\operatorname{\mathcal{QC}}$ is idempotent complete. More generally, for every uncountable cardinal $\kappa$, the $\infty$-category $\operatorname{\mathcal{QC}}^{< \kappa }$ of $\kappa$-small $\infty$-categories is idempotent complete. To prove this, we can use Propositions 8.4.4.6 and 8.4.1.12 to reduce to the case where $\kappa$ has uncountable cofinality. In this case, the $\infty$-category $\operatorname{\mathcal{QC}}^{< \kappa }$ admits sequential colimits (Example 7.6.7.7), so the desired result follows from Corollary 8.4.4.17.

Example 8.4.4.20. Let $\operatorname{\mathcal{S}}$ denote the $\infty$-category of spaces. Then $\operatorname{\mathcal{S}}$ is idempotent complete. More generally, for every uncountable cardinal $\kappa$, the $\infty$-category $\operatorname{\mathcal{S}}^{< \kappa }$ of $\kappa$-small spaces is idempotent complete. This follows from Example 8.4.4.19 and Proposition 8.4.4.6, since the full subcategory $\operatorname{\mathcal{S}}^{< \kappa } \subseteq \operatorname{\mathcal{QC}}^{< \kappa }$ is closed under the formation of retracts (Remark 8.4.1.13).

Warning 8.4.4.21. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. If $\operatorname{\mathcal{C}}$ is idempotent complete, then its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be idempotent complete. For example, the $\infty$-category of spaces $\operatorname{\mathcal{S}}$ is idempotent complete (Example 8.4.4.20), but its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ is not (see Proposition 8.4.7.15).


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# Homotopy of Spectra Maps Induced by Homotopy of Functions

So, I'm still working through Adams' lecture notes, and here's something that I haven't been able to immediately suss out:

It is clear that for $F$ a spectrum, $K$ a finite CW-complex and $E$ its associated suspension spectrum, a function $f:\Sigma^nK\to F_n$ induces a map of spectra (since we can just take everything above $\Sigma^nK$ to be the cofinal subspectrum and the suspensions induce all the higher maps).

Now, we are ultimately going to be considering the object $A=\mathrm{colim}_n[\Sigma^nK,F_n]$. So, suppose two maps $f:\Sigma^n\to F_n$ and $g:\Sigma^m\to F_m$ induce the same object in $A$. Then they must (by the nature of the directed system we index our colimit over) coincide at some finite point, i.e. for some $p$ the maps $\Sigma^pK\overset{\Sigma^{p-n}f}\to\Sigma^{p-n}K\to F_p$ and $\Sigma^pK\overset{\Sigma^{p-n}g}\to\Sigma^{p-n}K\to F_p$ are homotopic or define the same object in $[\Sigma^pK,F_p]$.

Adams' next claim is that this homotopy can be extended to the entire spectra (on suitable cofinal subspectra of course) and perhaps it's just my lack of expertise with suspensions and stuff, but I can't quite get it to work out. Something like, if we have a homotopy at a given level then we can just suspend that homotopy to get it on every higher level (i.e. a cofinal subspectrum).

Thanks! -Jon

-

If we have a homotopy at one level, that means that there is a map $h:\Sigma^pK\times I\to F_p$. Since this homotopy is pointed, it is constant on the base point, so we might as well consider it as a map $h:\Sigma^pK\wedge I^+\to F_p$. Thus, we have the next map $\Sigma h:\Sigma(\Sigma^pK\wedge I^+)\cong \Sigma^{p+1}K\wedge I^+\to F^{p+1}$ and so on all the way up, which is exactly what we need!


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# Jacobian transpose vs pseudoinverse

Let’s derive both celebrated algorithms from first order principles to see how they differ from each other.

Jacobian transpose. Let $x_d$ be the desired state of the end-effector, $x(q)$ its current state corresponding to the joint configuration $q$, and let $J$ denote the Jacobian $\partial x / \partial q$. Naive minimization of the error in the Cartesian space

where $\alpha$ is the step size.
Jacobian pseudoinverse. Consider a small change in the joint configuration $\Delta q$. It will lead to a small change in the state of the end-effector $\Delta x = J \Delta q$. Since we want the end-effector to move in the direction of $x_d$, let’s assume that $\Delta x = \alpha (x_d - x)$ with some positive $\alpha$. Question: “What is the smallest $\Delta q$ that accomplishes that?” It is well-known that the minimum norm solution to an underdetermined system of linear equations is given by the pseudoinverse. Thus, we arrive at the following gradient descent algorithm:
where $J^{+} = J^T (JJ^T)^{-1}$ if $J$ has full row rank (which is ususally the case).
Comparison. Note that both algorithms have the form $\Delta q = J^T \Delta x$. The difference between them is in the choice of $\Delta x$. Jacobian transpose algorithm sets the step in the Cartesian space proportional to the distance to the goal $\Delta x = \alpha (x_d - x(q))$, whereas Jacobian pseudoinverse accounts for the curvature of the space in addition, $\Delta x = (JJ^T)^{-1} \alpha (x_d - x(q))$.