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differential equations | Why differential equations? | https://physics.stackexchange.com/questions/349226/why-differential-equations | <p>Natural phenomena (e.g. heat flow) and systems (e.g. electrical circuits) are usually described using differential equations. Why is that?</p>
<p>Also, usually people use "<em>constant</em> coefficients <em>linear</em> differential equations" of low order (one or two, rarely three). Is this use (constant c... | <p>Given that time and space are believed to be continuous one would expect that the equations governing changes in time and space would reflect this continuity.</p>
<p>In other words, we can make sense of the concept of two points arbitrarily close in space or two moments arbitrarily close in time, something that dif... | 1,000 |
differential equations | Differential Equations for Physicists | https://physics.stackexchange.com/questions/455312/differential-equations-for-physicists | <p>I find differential equations in physics to be quite challenging so I'm looking for a book to help me master them.</p>
<p>I'm familiar with solving ordinary differential equations via seperation of variables but haven't really gone much further than that.</p>
<p>I was thinking about buying this: <a href="https://w... | <p>Are you a self-learning person? Any university course of math and physics for physicists suffice to cope with your difficulties. However, and I can say it from my professional experience, the learning never ends. So be ready to learn from different sources, points of view, etc., etc.</p>
| 1,001 |
differential equations | Accuracy of differential equations | https://physics.stackexchange.com/questions/178916/accuracy-of-differential-equations | <p>We use differential equations to model the world around us. For example, the logistic differential equation
$$\frac{dx}{dt} = rP\left(1-\frac PK\right)$$</p>
<p>is used to model population. However, it doesn't take into account things like climate, natural disasters, competition among other species, etc. Equations ... | <blockquote>
<p>How accurate are differential equations really, and to what accuracy can we predict future circumstances and events from them?</p>
</blockquote>
<p>Why does it matter that they are "differential equations"? Differential equations are just one type of model. The question is how accurate are theoretica... | 1,002 |
differential equations | Decouple differential equations | https://physics.stackexchange.com/questions/374895/decouple-differential-equations | <p>I have a system of two Second Order differential equations
$$r^2\ddot{r}−r^3(\dot{\varphi}^2+ω^2)=−GM$$
$$r \ddot{\varphi}+2 \dot{r}(\dot{\varphi}+\omega)=0 $$
using the conserved quantity $(\dot{\varphi}+\omega)r^2$, call it <em>Ω</em>.<br>
I have shown that it is indeed a conserved quantity, as its time-derivative... | <p>Since $h:=r^2(\dot{\varphi}+\omega)$ is conserved, $\frac{d}{dt}=(\frac{h}{r^2}-\omega)\frac{d}{d\varphi}$ and $\dot{r}=-h\frac{du}{d\varphi}$ with $u:=\frac{1}{r}+\frac{\omega r}{h}$ so $\ddot{r}=-h(\frac{h}{r^2}-\omega)\frac{d^2 u}{d\varphi^2}$. You'll want to rewrite your equations of motion in the coordinate sys... | 1,003 |
differential equations | Applications of delay differential equations | https://physics.stackexchange.com/questions/27143/applications-of-delay-differential-equations | <p>Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations).</p>
<p>It is clear that</p>
<ul>
<li>in biological (population) models usually pregnancy introd... | <p>In my corner of things what comes to mind is a recent paper by Atiyah and Moore <a href="https://arxiv.org/abs/1009.3176" rel="nofollow noreferrer">A Shifted View of Fundamental Physics</a>.</p>
| 1,004 |
differential equations | Recurrence differential equations | https://physics.stackexchange.com/questions/180701/recurrence-differential-equations | <p>We all know recurrence equations like e.q. Fibonacci relation</p>
<p>$$F_{n} = F_{n-1} + F_{n+1}$$</p>
<p>In order to find general expression for any $n$, we can use <em>generating function</em> method </p>
<p>$$G(x) = \sum\limits_{n=0}^{\infty}F_{n}x^{n}$$</p>
<p>or its variation <a href="http://en.wikipedia.or... | <p>I don't think that you need some complicated apparatus. You can simply solve this problem by writing </p>
<p>$$
\dot{c}_k(t) = M_{kj} c_j(t).
$$</p>
<p>Now because $M$ is symmetric and real you can find a transformation $\tilde{c}_k$ of the $c_k$ that diagonalizes $M$ and leads to trivial differential equations fo... | 1,005 |
differential equations | Physics and Linear Differential Equations | https://physics.stackexchange.com/questions/95795/physics-and-linear-differential-equations | <p>Why in physics, most of the physical systems are modelled by linear differential
equations?</p>
| <p>I think your qualification of "most" systems needs some clarification because really almost all of the classical universe is described by second-order, nonlinear partial differential equations. Fluids/liquids/gases and solids are described by the same set of second-order, nonlinear PDE's.</p>
<p>Linear equations, b... | 1,006 |
differential equations | Phase-amplitude stochastic differential equations | https://physics.stackexchange.com/questions/757179/phase-amplitude-stochastic-differential-equations | <p>In the book of <span class="math-container">$\textit{The Quantum World of Ultra-Cold Atoms and Light: Book 1 Foundations of Quantum Optics}$</span> by Peter Zoller and Crispin Gardiner on page 75, they derive the phase-amplitude stochastic differential equation for a thermalized oscillator.</p>
<p>From a complex Orn... | <p>Using the fact that:
<span class="math-container">$$
a=|\alpha|
$$</span>
You can use it to calculate:
<span class="math-container">$$
\langle a(t_1)a(t_2)\rangle =\langle |\alpha(t_1)\alpha(t_2)|\rangle
$$</span>
You can calculate in very special cases, but there is no simple general formula. Perhaps what is more r... | 1,007 |
differential equations | Monodromy matrix and differential equations | https://physics.stackexchange.com/questions/238521/monodromy-matrix-and-differential-equations | <p>What is the significance of monodromy matrix in the context of differential equations? I have seen some papers(<a href="http://arxiv.org/abs/1303.6955" rel="nofollow">1</a>,<a href="http://arxiv.org/abs/1403.6829" rel="nofollow">2</a>,<a href="http://arxiv.org/abs/1510.06685" rel="nofollow">3</a> etc) in CFT which u... | <p>I'm going to explain how the monodromy approach to computation of semi-classical conformal block arises.</p>
<p>The goal is to compute the conformal block corresponding to the exchange of operator $\mathcal{O}$ in the four-point function
$$
\langle V_1(z_1)V_2(z_2)V_3(z_3)V_4(z_4)\rangle,\qquad (1)
$$
in $V_1V_2-V_... | 1,008 |
differential equations | Simulating Interactions in QFT without differential equations | https://physics.stackexchange.com/questions/661999/simulating-interactions-in-qft-without-differential-equations | <p>As I understand it in QFT interactions are generally modeled as being from the exchange of virtual particles. If I was to think of how to simulate a classical analog I would model two spheres A and B, that each can only change velocity by emitting or absorbing an exchange sphere C. I would use a random number gene... | <p>There's no QFT without Green functions and much more, but something like your question is addressed by Mattuck's "drunken man propagator", followed by "the classical quasi particle propagator", in the introductory chapters of his book "A Guide To Feynman Diagrams In The Many-Body Problem&quo... | 1,009 |
differential equations | Exact differential equations and holonomic constraints | https://physics.stackexchange.com/questions/370835/exact-differential-equations-and-holonomic-constraints | <p>I understand that if a constraint equation given on a differential form is exact, that means it is also holonomic since I can find a solution. But there are other types of differential equations, like separable and linear, such that I can find an equation in the form of a holonomic constraint. Why is it that being e... | <ol>
<li><p>Here I would like to mention the notion of a <a href="https://www.google.com/search?as_epq=semi+holonomic+constraint" rel="nofollow noreferrer">semi-holonomic constraint</a>
$$ \sum_{j=1}^n a_j(q,t)~ \mathrm{d}q^j + a_t(q,t)~\mathrm{d}t~=~0, \tag{1'}$$
which puts an
<a href="https://en.wikipedia.org/wiki/I... | 1,010 |
differential equations | Using RC circuits to solve differential equations | https://physics.stackexchange.com/questions/319285/using-rc-circuits-to-solve-differential-equations | <p>As I was thinking about RC circuits it dawned upon me that under the correct configurations one could very efficiently solve differential equations by programming them into an RC circuit (the applications of this would be something like a very very fast hardware implementation of machine learning). </p>
<p>If you h... | <p>Consider this circuit:</p>
<p><a href="https://i.sstatic.net/RgbOF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/RgbOF.png" alt="enter image description here"></a></p>
<p>If the capacitor is initially charged, the system is governed by these equations:</p>
<p>$$\frac{{\rm d}v}{{\rm d}t} = \frac{-... | 1,011 |
differential equations | Solution to differential equation | https://physics.stackexchange.com/questions/643519/solution-to-differential-equation | <p>If I have a differential equations of the form <span class="math-container">$$\frac {d^2y}{dt^2}=\alpha^2y$$</span></p>
<p>Assuming the roots of the characteristic equation is complex the solution to the differential equation is: <span class="math-container">$$y=C_1e^{j\alpha t}+ C_2e^{-j\alpha t}$$</span> and after... | <p>There's two ways to look at it. The first one is as you said: we take the real part of the general solution. We can verify that if <span class="math-container">$\alpha^2$</span> is real, the real and imaginary parts of the general solution also happen to solve the differential equation on their own. So the real part... | 1,012 |
differential equations | Radioactive decay differential equations | https://physics.stackexchange.com/questions/550021/radioactive-decay-differential-equations | <p>I am trying to form a differential equation between two different isotopes, Uranium-238 and Thorium-234.
The rate of decay of an isotope is proportional to the amount present. So that:
<span class="math-container">$$
\frac{dx}{dt} = -kx
$$</span>
Where x is the amount of Uranium-238 and k is the constant if propor... | <p>The decay rate of the mother isotope <span class="math-container">$x$</span> depends only on the amount of <span class="math-container">$x$</span>, so that:</p>
<p><span class="math-container">$$\frac{\text{d}x(t)}{\text{d}t}=-kx(t)$$</span>
This solves easily:
<span class="math-container">$$\ln x=-kt +C$$</span>
I... | 1,013 |
differential equations | From differentials to differential equations | https://physics.stackexchange.com/questions/52547/from-differentials-to-differential-equations | <p>Suppose I have a function of time $t$ and position $(x,y)$ such that
\begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + p_y \,dy\end{equation}
where the subscript denotes a differentiation. In this case, I am able to derive a (partial) differential equation from this form.</p>
<p>I'd love to have your help to ... | <p>The $(dy)^2$ term is totally negligible, it's as if it was not there. If you had two differentials everywhere, then yes, it would lead to a 2nd order diff. equation.</p>
| 1,014 |
differential equations | Shadow method of solving differential equations | https://physics.stackexchange.com/questions/595945/shadow-method-of-solving-differential-equations | <p>While reading this answer by Rishab Navneet <a href="https://physics.stackexchange.com/a/595835/236734">here</a>, it is shown how we can visualize the harmonic oscillator as the shadow of a body moving in a circle onto a line. How was it found that the plane curve is a circle? More generally, is there a way to go fr... | <p>We have backtraced the curve for which SHM can be represented and if you solve the given differential equation
<span class="math-container">$$ \frac{d^2 \theta}{dt^2} = \frac{-g}{L} \sin \theta$$</span>
you will get a function and you can backtrace if possible.
after finding the function of SHM we have related it wi... | 1,015 |
differential equations | Solving the differential equations for self-induction | https://physics.stackexchange.com/questions/356130/solving-the-differential-equations-for-self-induction | <p>In my physics class we learned the equations for self-induction. Our teacher also showed us the differential equations and gave us the solutions. Because we didn't have differential equations in our Maths class he only told us that, if we were interested in how to solve these equation, we should look up separation o... | <p>The first step is $\frac{dI}{dt}=\frac{U_0}{L}-\frac{R}{L}I$, so we have $\frac{dI}{1-\frac{R}{U_0}I}=\frac{U_0}{L}dt$.</p>
<p>I hope you can continue now, is just integrate </p>
| 1,016 |
differential equations | Significance of exact solutions to differential equations | https://physics.stackexchange.com/questions/513007/significance-of-exact-solutions-to-differential-equations | <p>What is the importance of finding new exact solutions to partial differential equations? I kindly need someone to convince me, since my PhD will be on that. </p>
| <p>If you have an analytical/exact (in contrast to some say "discretized" or similar numerical) solution, interpretation of the coefficients in this solution is much more straightforward than tackling a generic numerical solution. These coefficients of the exact solution can also be fitted to experiments to gain an und... | 1,017 |
differential equations | Differential equations of a fluid-mechanical system | https://physics.stackexchange.com/questions/294572/differential-equations-of-a-fluid-mechanical-system | <p>I'm trying to simulate a real system, in order to do so I have modelled a physical system (fluid-mechanical) that behaves similarly with some simplifications. The physical model in question is as follows:</p>
<p><a href="https://i.sstatic.net/CP1Sf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CP1S... | 1,018 | |
differential equations | Differential equations ball with air resistance | https://physics.stackexchange.com/questions/317293/differential-equations-ball-with-air-resistance | <p>I'm trying to find the equation for a ball thrown from the ground with an initial velocity. Are these differential equations correct? I solved these and set the integrating constant to $v_0cos(\theta)$ for $v_x$ and $v_0sin(\theta)$ for $v_y$ and integrated again to get the function of position. Is that the correct ... | <p>I assume that you are taking the drag force as having magnitude $k|{\bf v}|^2$? In that case the you need to resolve the components of the drag vector which points backwards along ${\bf v}$, and get
$$
m\dot v_x= -kv_x\sqrt{v_x^2+v_y^2},
$$
$$
m\dot v_y= -kv_y\sqrt{v_x^2+v_y^2}-g.
$$
These are essentially impossi... | 1,019 |
differential equations | Proving Kepler's 1st Law without differential equations | https://physics.stackexchange.com/questions/86435/proving-keplers-1st-law-without-differential-equations | <p>Is there a way to show that the motion of Earth around the Sun is elliptical (<a href="https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion" rel="nofollow noreferrer">Kepler's 1st law</a>) from Newton's laws without resorting to the use of differential equations of motion?</p>
| <p>Newton's original proof was in fact based on geometry (he hadn't invented calculus yet). Richard Feynman devised his own, simpler geometric proof for one of his famous lectures. You can find it in <em>Feynman's Lost Lecture</em>, by Goodstein & Goodstein, and in this article: <a href="https://tlakoba.w3.uvm.edu/... | 1,020 |
differential equations | Does the SUVAT equations of motion (Kinematics) come from some differential equation? | https://physics.stackexchange.com/questions/606669/does-the-suvat-equations-of-motion-kinematics-come-from-some-differential-equa | <p>Wikipedia says about the equations of motion that;</p>
<blockquote>
<p>"If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics."</p>
</blockquote>
<p>And</p>
<blockquote>
<p>"A differential equation of motion, usually ... | <p>Yes they do; the SUVAT equations are directly derived from the simple relations between acceleration, velocity and position:
<span class="math-container">\begin{equation}
a=\frac{dv}{dt}\\
v=\frac{ds}{dt}.
\end{equation}</span><br />
Starting from the first equation, we find
<span class="math-container">\begin{equat... | 1,021 |
differential equations | Gravity Differential Equations | https://physics.stackexchange.com/questions/710179/gravity-differential-equations | <p>I was just messing around with Newton's Law of gravitation, when I had the idea of converting Newton's Law into differential form (more or less like Maxwell's equations).</p>
<p>I did the following:</p>
<h5>#1 Divergence of the field:</h5>
<p><span class="math-container">$$
\iint_C {\mathbf g \cdot d\mathbf S} = -4\... | <p>Newtonian gravity does not have any forces that are not radially inward or outward, so if you are in pure Newtonian gravity, there is no generalization of magnetism.</p>
<p>Now, if you've seen a <a href="https://physics.stackexchange.com/questions/64703/how-special-relativity-causes-magnetism">fancy derivation of ma... | 1,022 |
differential equations | Building realistic simulation using differential equations | https://physics.stackexchange.com/questions/519857/building-realistic-simulation-using-differential-equations | <p>I am building simulation using differential equations to model the motion of a damped vertical spring-mass system. I wish to use this simulation to extract data. For example, I am trying to find the effect of mass on the damping. </p>
<p>The problem I am facing is every time I run the model, I receive the same numb... | <p>The answer to the question as asked is <strong>“you add randomness to the system parameters and initial conditions”</strong>. Some search terms are “uncertainty quantification” and “design of experiments”.</p>
<p>But ... those techniques are used for models where there isn’t a detailed analytic understanding of the... | 1,023 |
differential equations | Looking for a good book on Differential Equations | https://physics.stackexchange.com/questions/571726/looking-for-a-good-book-on-differential-equations | <p>I know many of you are tired of book recommendation posts and questions. But I am self learning Theoretical Physics, and I am having a hard time choosing a book to learn differential equations (ODEs). I really want a good understanding of differential equations; I have been told that ODEs and PDEs are the language o... | <p>Welcome here!</p>
<p>If you are just starting you may want to use a very didactic textbook:</p>
<p>Zill. - Differential Equations with Boundary Value Problems (look for the solution's manual, it helps if you are an autodidact)</p>
<p>Then a very good one to get into the maths of physics I'd recommend this one:</p>
<... | 1,024 |
differential equations | Differential Equations for Block Diagram of Satellite Attitude Control System | https://physics.stackexchange.com/questions/122219/differential-equations-for-block-diagram-of-satellite-attitude-control-system | <p><img src="https://i.sstatic.net/zuU5y.png" alt="Text Book Cut Out"></p>
<p>I am trying to understand the procedure to setup differential equations from a block diagram. The enclosed example is about the attitude control of a satellite. The ultimate goal is to find a state-space system representation of the model. T... | <p>I think I've understood the basic tricks that the book used, which are what are tripping you up.</p>
<p>The main difficulty I see you struggling with is the arbitraryness of the variable selection. <em>Could</em> you write the state-space system differently? Yes. There are many ways you could write it. The part... | 1,025 |
differential equations | Research problems in application of Lie groups to differential equations | https://physics.stackexchange.com/questions/100800/research-problems-in-application-of-lie-groups-to-differential-equations | <p>Are there any open problems in physics involving Lie groups and differential equations for a phd theses. </p>
<p>Some applications are say, Noether's theorem in classical or quantum field theory. But I am not sure if those topics lead to any research problems. </p>
<p>So any idea about prospective research problem... | <p>I do not believe that there are any.<br>
You can check Stephanie Singer's book on Lie groups as applied as symmetries of differential equations, and also the book on mechanics and look at the unfortunately old-fashioned review
<a href="http://people.ucsc.edu/~rmont/papers/Symm_in_Mech_Review.PDF" rel="nofollow">htt... | 1,026 |
differential equations | Solving differential equations without approximations? | https://physics.stackexchange.com/questions/133974/solving-differential-equations-without-approximations | <p>In physics, many problems start with a mathematical relationship of the physical phenomenon at hand, and then, in many occasion, always only leave whatever in the first order to get a nice and solvable differential equation. Then there may be terms of higher order considered later, by as far as I know, it rares goes... | <p>This is a quite general question. Whether or not one should use an approximation depends on several things. Disclaimer: my answer is not restricted to differential equations and contains examples from perturbation theory, but the general idea still applies. </p>
<p>The most important question is whether an approxi... | 1,027 |
differential equations | How to make sense of quantum fields differential equations? | https://physics.stackexchange.com/questions/320114/how-to-make-sense-of-quantum-fields-differential-equations | <p>A quantum field is an operator valued function, that is, a function $\varphi(x)$ defined on spacetime which assigns operators on a Hilbert space to each event $x$. In a more rigorous approach a quantum field could be defined as an operator valued distribution on spacetime.</p>
<p>Anyway, it is quite common that the... | <p>The fields of a QFT are not functions of the spatial coordinates $\boldsymbol x\in\mathbb R^n$, but operator-valued distributions (borrowing Wightman's terminology). The notion of the fields being functions of time ("sharp-time fields") can be kept in general, but their dependence on $\boldsymbol x$ is "too singular... | 1,028 |
differential equations | Higher-order derivatives than second-order differential equations | https://physics.stackexchange.com/questions/679352/higher-order-derivatives-than-second-order-differential-equations | <p>From <a href="https://doi.org/10.1063/1.2155755" rel="nofollow noreferrer">https://doi.org/10.1063/1.2155755</a></p>
<blockquote>
<p>he limited himself to second-order differential equations.</p>
</blockquote>
<blockquote>
<p>Our experience in elementary-particle physics has taught us that any term in the field equa... | <ol>
<li><p>OP is right that if the Lagrangian density remains of 1st order, then the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation" rel="nofollow noreferrer">Euler-Lagrange (EL) equations</a> will only be of 2nd order. See also e.g. <a href="https://physics.stackexchange.com/q/18588/2451">this... | 1,029 |
differential equations | On different methods of solving differential equations | https://physics.stackexchange.com/questions/778559/on-different-methods-of-solving-differential-equations | <p>I've studied the basic concepts of partial differential equations, and one question comes to my mind. What are the propuse of the diferent methods of resolution of differential equations. For example if you start with:
<span class="math-container">$$
\frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} - \frac{\partial... | <p>I do not understand what you mean when you say that the solutions you get with separation of variables are stationary, since they depend explicitly on time. Maybe it will help you to realize that your function
<span class="math-container">\begin{equation}
\Psi_n(t,x) = \left( A_n \sin(k_n x) + B_n \cos(k_n x) \righ... | 1,030 |
differential equations | Application for differential equation of higher order | https://physics.stackexchange.com/questions/283292/application-for-differential-equation-of-higher-order | <p>We found some interesting insights in differential equations of the form</p>
<p>$y^{(n)}(x)+F_\lambda(y(x),y'(x),...,y^{(n-1)}(x))=0$,</p>
<p>i.e. for ordinary differential equations of $n$-th order with $n\geq2$. The function $F$ is polynomial which can include a set of parameters $\lambda$.</p>
<p>We know, that... | <p>That's just not true. If a linear system has $n$ independent ways in which energy can be stored as states, and energy can flow between these states, then you can model the system with an nth order polynomial.</p>
<p>Granted some systems can be approximated by a linear 2nd order rational polynomial function, but a c... | 1,031 |
differential equations | Differential equation | https://physics.stackexchange.com/questions/575708/differential-equation | <p>I am trying to solve the following differential equation;</p>
<p><span class="math-container">$$\frac{d^2 x}{d t^2}=-\omega^2 x \delta(t-t^\prime).$$</span></p>
<p>I know this is of the form</p>
<p><span class="math-container">$$x(t)= A \sin(\omega t) + B \cos(\omega t).$$</span></p>
<p>However this delta Dirac func... | <p>You need to find the solutions for <span class="math-container">$t> t'$</span> and for <span class="math-container">$t> t'$</span> and then impose the integration constants by the boundary conditions. The problem is actually very similar to that of a delta-function barrier in quantum mechanics, but with zero e... | 1,032 |
differential equations | Why are differential equations used a lot in physics? | https://physics.stackexchange.com/questions/733203/why-are-differential-equations-used-a-lot-in-physics | <p>I have heard from my physics teacher that differential equations are very useful in physics. In what parts of physics exactly is it useful? Why are they generally useful?</p>
| <p>I'd like to elaborate on an earlier answer.</p>
<p>In general the quantities that go into the equations come in chuncks, related by differentiation. The most known example is the trio: position, velocity, acceleration. Velocity is the first time derivative of position, acceleration being the second time derivative.<... | 1,033 |
differential equations | Complex exponential method of solving differential equations | https://physics.stackexchange.com/questions/623561/complex-exponential-method-of-solving-differential-equations | <p>In the <a href="https://www.feynmanlectures.caltech.edu/I_23.html" rel="nofollow noreferrer">twenty third Feynman lecture</a>, the solution of the following differential equation is discussed:</p>
<p><span class="math-container">$$ \frac{d^2 x}{dt^2} + \frac{kx}{m} = \frac{F}{m}$$</span></p>
<p>AFter 'complexifying'... | <p><span class="math-container">$x$</span> and <span class="math-container">$F$</span> can each be expressed as a Fourier integral:
<span class="math-container">$$x(t)=\int x(\omega)e^{i\omega t} d\omega$$</span>
<span class="math-container">$$F(t)=\int F(\omega)e^{i\omega t} d\omega$$</span>
This of course assumes tha... | 1,034 |
differential equations | Numerical solution of differential equations, e.g. the three-body problem | https://physics.stackexchange.com/questions/815292/numerical-solution-of-differential-equations-e-g-the-three-body-problem | <p>What forms of differential equations have numerical solutions with errors that go to zero with sufficient computational power? For example, suppose I want to solve a differential equation <span class="math-container">$E$</span> for a position vector <span class="math-container">$r$</span> at time <span class="math-c... | <p>Any algorithm for the numerical solution of ordinary differential equations provides a discretized approximation of the exact solution and should guarantee that the global error after a finite time <span class="math-container">$T$</span> should vanish by refining the discretization. The quality of different algorith... | 1,035 |
differential equations | Hamiltonian from a differential equation | https://physics.stackexchange.com/questions/249567/hamiltonian-from-a-differential-equation | <p>In my differential equations course an example is given from the Lotka-Volterra system of equations:</p>
<p>$$ x'=x-xy$$</p>
<p>$$y'=-\gamma y+xy.\tag{1}$$</p>
<p>This is then transformed by the substitution: $q=\ln x, p=\ln y$. </p>
<p>$$ q'=1-e^p$$</p>
<p>$$p'=-\gamma +e^q.\tag{2}$$</p>
<p>Then without any e... | <p>This is explained in part II of my Phys.SE answer <a href="https://physics.stackexchange.com/a/53637/2451">here</a>, which shows that a 2D system always has a Hamiltonian description locally.</p>
<p>It turns out, that before the non-canonical transformation $(x,y) \to (q,p)$, from the first pair of eoms (1) alone, ... | 1,036 |
differential equations | Tensor differential equation | https://physics.stackexchange.com/questions/822202/tensor-differential-equation | <p>How to solve the following differential equation with tensor indices?</p>
<p><span class="math-container">$\epsilon_{\mu\nu}\partial^{\gamma}\partial_{\gamma}f-2i\epsilon_{\mu\nu}p.\partial f+ip_{\mu}x^{\gamma}\epsilon_{\nu\gamma}+ip_{\nu}x^{\gamma}\epsilon_{\mu\gamma}-i(p.x)\epsilon_{\mu\nu}+2\epsilon_{\mu\nu}=0$</... | 1,037 | |
differential equations | Decoupling coupled differential equations in dynamically coupled two state system | https://physics.stackexchange.com/questions/252692/decoupling-coupled-differential-equations-in-dynamically-coupled-two-state-syste | <p>Consider the following dynamically coupled two state hamiltonian, $$H=-B\sigma_z-V(t)\sigma_x.$$Taking the eigenfunctions of $\sigma_z$ ($|+>$ and $|- >$) as basis vectors, we have the wave function to be $$\Phi=c_
1|+>+ c_2|->$$ and we get coupled differential equations for the time evolution of these t... | <p>The system can be separated, but not necessarily in nice form. For instance, the time derivative of the first eq. reads
$$
i\hbar {\ddot c}_1 = - B {\dot c}_1 - {\dot V}c_2 - V {\dot c}_2
$$
Now remove $c_2$ using again the first eq.,
$$
c_2 = -\frac{i\hbar}{V} {\dot c}_1 - \frac{B}{V} c_1
$$
and ${\dot c}_2$ usin... | 1,038 |
differential equations | Differential equation in non-uniform circular motion | https://physics.stackexchange.com/questions/359220/differential-equation-in-non-uniform-circular-motion | <p>I have a question which states</p>
<blockquote>
<p>An astronaut is conducting an experiment on a spaceship under conditions of zero gravity. A bead is threaded on a circular wire, and set in motion with angular velocity $ \omega _0 $ about the centre. If the coefficient of friction between the bead and the wire i... | <p>I have solved the equation by doing it in two steps. First, rather than thinking of it as $\ddot{\theta} = -\mu \dot{\theta} ^2$, treat it as $\frac{d\omega}{dt} = -\mu \omega ^2$, which can be solved by separating the variables:
$$ \begin{align} \int \frac{1}{\omega ^2} \ d\omega &= \int -\mu \ dt \\ -\frac{1}{... | 1,039 |
differential equations | How do I know which equations can be treated as differential equations and which can't? | https://physics.stackexchange.com/questions/614395/how-do-i-know-which-equations-can-be-treated-as-differential-equations-and-which | <p>I'm sometimes mystified by the use of differentials in physics. I don't understand which formulas—on which occasions—can be thought of as differential equations and which cannot.</p>
<p>While discussing work done by a piston during an isothermal process, my textbook does not treat <span class="math-container">$PV=nR... | <p><span class="math-container">$$PV=nRT\tag{1}$$</span></p>
<p>can not be considered a differential equation, simply because it <strong>contains no differentials</strong>.</p>
<p>Now, you can't just go and differentiate that equation as:</p>
<p><span class="math-container">$$P\mathrm{d}V=nR\mathrm{d}T$$</span></p>
<p>... | 1,040 |
differential equations | Rewriting the Hydrogen Schrodinger Equation as a system of differential equations | https://physics.stackexchange.com/questions/141238/rewriting-the-hydrogen-schrodinger-equation-as-a-system-of-differential-equation | <p>I have only ever seen the Schrodinger equation for the hydrogen atom written out in a form like this:
$$
-\frac{\hbar^2}{2\mu}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial\psi}... | <p>If you assume <a href="http://en.wikipedia.org/wiki/Separation_of_variables" rel="nofollow">separability</a> of the wave function, i.e., $\psi(\mathbf x)=u(x)v(y)w(z)$, you can solve the individual components separately:
\begin{align}
-\frac{\hbar^2}{2\mu}\frac{d^2u(x)}{dx^2}+V_1(x)u(x)&=E_1u(x)\\
-\frac{\hbar^2... | 1,041 |
differential equations | Relativity and differential equation | https://physics.stackexchange.com/questions/488335/relativity-and-differential-equation | <p>I have a question regarding Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation ISBN 978-0-7167-0344-0. It is a book about Einstein's theory of gravitation.</p>
<p>In page 166 of chapter 6.2 about Hyperbolic Motion, the authors present a person feeling constant acceleration <span class="... | <p>solution:</p>
<p><span class="math-container">$${\frac {d}{d\tau}}u_{{0}} \left( \tau \right) -gu_{{1}} \left( \tau
\right) =0\tag 1
$$</span></p>
<p><span class="math-container">$${\frac {d}{d\tau}}u_{{1}} \left( \tau \right) -gu_{{0}} \left( \tau
\right) =0\tag 2
$$</span></p>
<p>and the constraint condition ... | 1,042 |
differential equations | Multiple time dimensions and understanding ultrahyperbolic differential equations | https://physics.stackexchange.com/questions/836427/multiple-time-dimensions-and-understanding-ultrahyperbolic-differential-equation | <p>On article "On the dimensionality of spacetime (<a href="https://space.mit.edu/home/tegmark/dimensions.pdf" rel="nofollow noreferrer">https://space.mit.edu/home/tegmark/dimensions.pdf</a>) Max Tegmark writes about ultrahyperbolic differential equations leading to unpredictability:</p>
<blockquote>
<p>"If a... | 1,043 | |
differential equations | What is the partial differential equation expansion of the Einstein Field Equations? | https://physics.stackexchange.com/questions/189515/what-is-the-partial-differential-equation-expansion-of-the-einstein-field-equati | <p>I have read that the Einstein Field Equations (<a href="http://en.wikipedia.org/wiki/Einstein_field_equations" rel="nofollow">http://en.wikipedia.org/wiki/Einstein_field_equations</a>) can be expressed as a series of differential equations. Some say 16, others say 10 (The disparity seems to stem from a simplificati... | <p>As asked in the comments, here is one answer : </p>
<p>One formalism where it is somewhat common to expand the Einstein equations into a full set of equations is the Newman-Penrose formalism. Not quite common as it uses both spinors instead of tensors and the coordinates are weird complex null-vectors, but it shoul... | 1,044 |
differential equations | How second-order differential equations do not violate causality? | https://physics.stackexchange.com/questions/323233/how-second-order-differential-equations-do-not-violate-causality | <p>The second order differential equations are time reversible. That means: they don't distinguish the time arrow direction. There is no reason for the time to flow forward. </p>
<p>My professor told me that there are two solutions to such equations, one of which describes processes going forward and one backward in t... | <p>Causality is not a hard-science topic as much as it is a philosophy of science topic. Causality is actually a <em>huge</em> issue in philosophy because, while typically want to say causality exists, it's actually <em>markedly</em> difficult to pen a description of it in a language which can stand up to the rigors o... | 1,045 |
differential equations | Differential equation for an accelerometer | https://physics.stackexchange.com/questions/317718/differential-equation-for-an-accelerometer | <p>I am having troubles deriving the 2nd order differential equation for the system below, where $r=y-s$. According to my lecture notes the differential equation is</p>
<p>$$
M\frac{d^2r}{dt^2}+b\frac{dr}{dt}+kr=-M\frac{d^2s}{dt^2}=-Ma \\
\ddot r+2\zeta \omega_0 \dot r+\omega_0² r=-a,
$$</p>
<p>whereas $ \omega_0 = ... | 1,046 | |
differential equations | General question about making differential equations dimensionless | https://physics.stackexchange.com/questions/446939/general-question-about-making-differential-equations-dimensionless | <p>Suppose you have a set of differential equations that you wish to normalize/make dimensionless. From what I've seen, you can usually use dimensional analysis to figure out a good choice of constants to make your variables and parameters dimensionless. However, in fluid dynamics, for example, you also have a few dime... | 1,047 | |
differential equations | Differential equations of a forced coupled spring-pendulum system | https://physics.stackexchange.com/questions/615333/differential-equations-of-a-forced-coupled-spring-pendulum-system | <p>Currently working on a problem and I can really figure out how to write the differential equations for it. Here's the situation:</p>
<p><a href="https://i.sstatic.net/O1WAM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/O1WAM.png" alt="Image of the system" /></a></p>
<p>So we have a mass <span class=... | 1,048 | |
differential equations | Kerr geodesics differential equations in equatorial plane | https://physics.stackexchange.com/questions/43629/kerr-geodesics-differential-equations-in-equatorial-plane | <p>With friend, we are writing an interactive educational simulation of particle falling into a black hole. </p>
<p>Currently we use <a href="http://en.wikipedia.org/wiki/Schwarzschild_geodesics#Geodesic_equation">Schwarzschild geodesics</a>. However, we want to generalize it to the case of <a href="http://en.wikipedi... | <p>I'll follow <em>Gravitation</em> by Misner, Thorne, and Wheeler (hereafter MTW), which is the standard <strike>reference textbook</strike> encyclopedic tome for the field despite its age.</p>
<p>Let $\lambda$ parametrize the path such that the derivative with respect to it gives the 4-momentum. Using Boyer-Lindquis... | 1,049 |
differential equations | Schwarzschild metric system of geodesic differential equations | https://physics.stackexchange.com/questions/675913/schwarzschild-metric-system-of-geodesic-differential-equations | <p>Suppose that we are given the Schwarzschild metric and its Lagrangian <span class="math-container">$L=-(1-\frac{R}{r})t'^2 + (1-\frac{R}{r})^{-1}r'^2+r^2 \theta'^2+r^2 \sin^2(\theta)\phi'^2$</span> where R=<span class="math-container">$r_s=2GM$</span> and <span class="math-container">$x'=\frac{d}{d \tau}$</span> <s... | <p>First, it is most definitely not necessary to transform your second order differential equations into first order differential equations in order to solve them. However, if you wish to do so for whatever reason, then the systematic way to do that is by using the Hamiltonian approach.</p>
<p>Here we define the moment... | 1,050 |
differential equations | Wave propagation speed in non-linear differential equations | https://physics.stackexchange.com/questions/750453/wave-propagation-speed-in-non-linear-differential-equations | <p>Could it happen than a solitary travelling wave (soliton) had a different propagation speed when seen from the usual wave equations from that in a non-linear equation. I mean, suppose a solution <span class="math-container">$F=f(x-vt)+g(x+vt)$</span> of the usual wave equation.</p>
<p>Could it happen than the "... | 1,051 | |
differential equations | Solution to pendulum differential equation | https://physics.stackexchange.com/questions/653845/solution-to-pendulum-differential-equation | <p>In a chapter on oscillations in <a href="https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums" rel="noreferrer">a physics book</a>, the differential equation <span class="math-container">$$\ddot{\theta}=-\frac{g}{L}\sin(\theta)$$</span> is found and solved using the small-angle-approximation <... | <p>The pendulum problem can be solved exactly if an elliptic integral is used.</p>
<p>The elliptic integral in question is defined via
<span class="math-container">\begin{equation}
F(\phi,k)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}}\, .
\end{equation}</span>
This integral originated when mathematicians investig... | 1,052 |
differential equations | Hypergeometric Function: Differential Equation | https://physics.stackexchange.com/questions/299251/hypergeometric-function-differential-equation | <p>In Birrel & Davies: <em>QFT in curved spacetime</em> it is written that the following differential equation can be solved in terms of hypergeometric functions.
$$(\partial_t^2 +(k^2+c(t)m^2))\phi(t)=0.$$
But there is no reference and no method listen.
Could somebody please help me solve this equation for $c(t)=(... | <p>This example in Birrell & Davies is quite tricky and in order to get the exact answer given, you need to manipulate the differential equation and solve it by hand as far as you can get.
This involves quite a bit of algebraic manipulation and properties of the hypergeometric functions, but the outline is this.</p... | 1,053 |
differential equations | Differential Equations - Waves (Physics self-study suggestions) | https://physics.stackexchange.com/questions/75506/differential-equations-waves-physics-self-study-suggestions | <p>I apologize ahead of time, in case this post is not allowed. </p>
<p>After taking a few courses at a community college, I've taken the fall 2013 semester off (I was accepted into a university for the spring 2014 semester). I'm really looking to spend the next 5 months on concentrated self-study to be a bit ahead of... | 1,054 | |
differential equations | Why do we use differential equations in physics instead of $h$-difference ones? | https://physics.stackexchange.com/questions/369481/why-do-we-use-differential-equations-in-physics-instead-of-h-difference-ones | <p>Since we don't know whether space and time are discrete or continuous wouldn't it be a better idea to use $h$-difference equations where the derivative is $$f'(x) =\frac{f(x+h)-f(x)}{h},$$ since they are more general and by sending $h$ to 0, we would have the usual differential equations. So why do we prefer differe... | <p>We can name a lot of reasons why one should prefer <em>differentials</em> to <em>finite differences</em>, but I guess one of the most physical ones is the relativity!</p>
<p>We assume that there is a finite speed limit for information transfer, that is the speed of light, hence any physical quantity should be local... | 1,055 |
differential equations | Solving the rocket differential equation | https://physics.stackexchange.com/questions/449027/solving-the-rocket-differential-equation | <p>I'm trying to derive the rocket equation.</p>
<p>I'm pretty sure that the differential equation for the rocket equation is</p>
<p><span class="math-container">$$v(t)\delta t =\frac{m(t)\delta t }{m(t)} V_e$$</span></p>
<p>where </p>
<ul>
<li><span class="math-container">$v(t)\delta t$</span> is the rate of chang... | <p>First of all, I believe part of the reason you're getting confused is that you're using confusing notation. Instead of <span class="math-container">$v(t) \delta t$</span>, the usual way of writing the rate of change of the rocket's velocity is <span class="math-container">$dv/dt$</span>; and for the rate of change ... | 1,056 |
differential equations | Constructing differential equation from arbitrary Hamiltonian | https://physics.stackexchange.com/questions/190164/constructing-differential-equation-from-arbitrary-hamiltonian | <p>Suppose I begin with the time-independent Schrodinger equation
$$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$
ordinarily we specify the function $V$ and then solve for a set of eigenfunctions and eigenvalues. And just to be slightly more general, we do the same thing with Sturm-Liouvil... | <p>After thinking about it, as long as the original eigenvalues are non-degenerate it should be possible to have the new Hamiltonian be represented by a differential equation of arbitrarily high order. The key is that the projection operators $P_n$ onto the eigenfunctions exist in the algebra generated by the original ... | 1,057 |
differential equations | Trouble Solving Partial Differential Equation | https://physics.stackexchange.com/questions/534832/trouble-solving-partial-differential-equation | <p>I'm solving the velocity profile of a fluid flow for a circular channel with an oscillating pressure gradient <span class="math-container">$\frac{dp}{dx}=\frac{\Delta p}{\rho L}e^{-i\omega t}$</span>. I plugged in to the Navier Stokes equations and am having trouble figuring out how to approach the solution to the p... | <p>I disagree with Chet, I think complex numbers are the way to go here.</p>
<p>As ever in these kind of problems, given that we have an oscillating pressure gradient, it makes sense to look for an oscillating velocity profile,
<span class="math-container">$$
u(r,t) = \hat{u}(r)e^{-i\omega t}.
$$</span>
Then your PDE... | 1,058 |
differential equations | Why must the field equations be differential? | https://physics.stackexchange.com/questions/13466/why-must-the-field-equations-be-differential | <p>In Landau–Lifshitz's <em>Course of Theoretical Physics</em>, Vol. 2 (‘Classical Fields Theory’), Ch. IV, § 27, there is an explanation why the field equations should be linear differential equations. It goes like this:</p>
<blockquote>
<p>Every solution of the field equations gives a field that can exist in nature. ... | <p>This does not really answer your question why the equation should be differential. But
I think that the two notions of locality you mentioned are just equivalent, if I am not mistaken.</p>
<p>Let us prove that the second definition impies the first one. One has to show that if a point $x$ does not belong to $supp(f... | 1,059 |
differential equations | Is the $Ψ$ in the Schrödinger Equation the same as the $Ψ$ in Exact Equations First Order Differential Equations? | https://physics.stackexchange.com/questions/416260/is-the-%ce%a8-in-the-schr%c3%b6dinger-equation-the-same-as-the-%ce%a8-in-exact-equations-fi | <p>The Schrödinger equations have the term $\Psi$, which is the wave function.</p>
<p><a href="http://scienceworld.wolfram.com/physics/SchroedingerEquation.html" rel="nofollow noreferrer">http://scienceworld.wolfram.com/physics/SchroedingerEquation.html</a></p>
<p>I do not know what type of equation the wave function... | <p>The Schrödinger equation is a partial differential equation. Its type depends on the Hamilton operator and the fact whether we have time-independent or time-dependent equation. In fact, it is not even strictly speaking a wave-equation in the <a href="https://en.wikipedia.org/wiki/Wave_equation" rel="noreferrer">math... | 1,060 |
differential equations | Applications of partial differential equations in material science | https://physics.stackexchange.com/questions/160332/applications-of-partial-differential-equations-in-material-science | <p>I've been asked to find a partial differential equation that has applications in material science. However we are not allowed to use the heat equation. I have found Fick's laws (basically the heat equation), and the Schrodinger equation, but I was wondering if there were any other prominent applications in material ... | <p>The <a href="http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" rel="nofollow">Navier-Stokes Equations</a> and its variants are some of the most important/relevant partial differential equations in material science.</p>
| 1,061 |
differential equations | Is resonance a general property of second-order differential equations? | https://physics.stackexchange.com/questions/749963/is-resonance-a-general-property-of-second-order-differential-equations | <p>I have read at this site as an answer at a question about how antennas work but that is not important</p>
<p>The resonant frequency of an antenna is determined by its constitution. Mathematically speaking, this is a general property of second order differential equations but in down-to-earth terms any AC circuit wit... | <p>Consider the second order differential equation
<span class="math-container">\begin{align}
f'' - \alpha^2 f = C \cos(\omega t)
\end{align}</span>
with <span class="math-container">$\alpha$</span> a real constant. (Note the sign of the second term, which makes this equation different from the equation of motion for a... | 1,062 |
differential equations | Modifying differential equations representing a projectile system to account for an arbitrary force | https://physics.stackexchange.com/questions/175197/modifying-differential-equations-representing-a-projectile-system-to-account-for | <p>The following series of differential equations represents a projectile's path when solved (g=9.81):</p>
<p><img src="https://i.sstatic.net/1gXIu.png" alt="The system"></p>
<p>Here is some sample output from this system (with initial values x,y=0, v=1500, theta=1.33):</p>
<p><img src="https://i.sstatic.net/26pdx.p... | <p>UPDATE: While I was typing the answer below, I see that you came to the same conclusion as I do in my answer below.</p>
<hr>
<p>From your $\dot \theta$ equation, it appears that a positive $x$ component of force acts to <em>increase</em> the rate of change of the angle when $0 \lt \theta \lt \pi$.</p>
<p>But, th... | 1,063 |
differential equations | Must multiple forces be expressed as a differential equation? | https://physics.stackexchange.com/questions/262473/must-multiple-forces-be-expressed-as-a-differential-equation | <p>This may be a stupidly obvious question, but can multiple forces (such as acceleration due to gravity and air resistance acting on a falling object) be expressed algebraicly or must it be written in the form of a differential equation? Since I don't know much about differential equations I have struggled to figure ... | <p>You only need differential equations when you are trying to find movements, and the forces change over time (or position).</p>
<p>At any given time, the forces can be written as a normal equation. Differential equations come into play when you look at their changing over time.</p>
| 1,064 |
differential equations | Why can you integrate with different bounds in thermal expansion differential equations? | https://physics.stackexchange.com/questions/814191/why-can-you-integrate-with-different-bounds-in-thermal-expansion-differential-eq | <p>I am just an independent student and was learning thermal expansion with differential equations and i saw someone on the internet solving the differential equation for the law like below:</p>
<p><span class="math-container">$$\frac{1}{L}\frac{dL}{dT}=\alpha$$</span>
<span class="math-container">$$\int_{L_{0}}^{L}\fr... | <p>It is recommended to use clearer notation such as the one Riley Scott Jacob has introduced, where the limits of integration are <span class="math-container">$T_A$</span> and <span class="math-container">$T_B$</span>, with <span class="math-container">$L_A=L(T_A)$</span> and <span class="math-container">$L_B = L(T_B)... | 1,065 |
differential equations | Vortex solution to Differential Equation | https://physics.stackexchange.com/questions/400029/vortex-solution-to-differential-equation | <p>I am looking for a differential equation whose solutions are what I call "open" vortices.
These vortices are not closed in themselves, but sort of "absorb" the surrounding "fluid" and also "emit" it.
I know that the Gross–Pitaevskii equation has vortices as solutions, but these are closed vortices as far as I know.<... | <p>It is not too hard to come up with dynamical systems with both attractive and repulsive fixed points. Here is a simple one: $$x'=(1-k)\sin(x)+k\sin(y)$$
$$y'=(1-k)\sin(y)-k\cos(x)$$ where $k$ is a mixing constant. Here is the vector field for $k=0.8$.</p>
<p><a href="https://i.sstatic.net/BiH0w.png" rel="nofollow n... | 1,066 |
differential equations | Lagrangian for two coupled second order linear differential equations | https://physics.stackexchange.com/questions/545343/lagrangian-for-two-coupled-second-order-linear-differential-equations | <p>Consider a system of two coupled linear differential equations
<span class="math-container">$$
\left(
\begin{bmatrix}
\Omega
\end{bmatrix}^{-1}
+ \frac{d^2}{dt^2} \right)
\vec{V}(t)
=
\begin{bmatrix}
C
\end{bmatrix}^{-1}
\vec{J}(t)
+ \begin{bmatrix}
\Omega
\end{bmatrix}^{-1} \vec{K}(t)
$$</span>
where <s... | <p><span class="math-container">$\boldsymbol{\S}$</span> <strong>A. A special case : symmetric</strong> <span class="math-container">$\Omega^{\boldsymbol{-}1}$</span></p>
<p>Let the <span class="math-container">$2\times2$</span> real symmetric matrices
<span class="math-container">\begin{equation}
C^{\boldsymbol{-}1}\b... | 1,067 |
differential equations | Numerical solution of two coupled second order differential equations of motion | https://physics.stackexchange.com/questions/100368/numerical-solution-of-two-coupled-second-order-differential-equations-of-motion | <p>Is there a numerical algorithm for solving a pair of coupled second order differential equations?</p>
<p>This question arises from a homework problem that I have that involves two dimensional projectile motion. The problem is as follows:</p>
<blockquote>
<p><em>An object is fired through a viscous fluid that has... | 1,068 | |
differential equations | Help recognizing partial differential equation | https://physics.stackexchange.com/questions/192341/help-recognizing-partial-differential-equation | <p>I would be very grateful if someone could tell me something about the following partial differential equation:</p>
<p>$$
\frac{\partial U}{\partial t} = K * (\frac{\partial^2 U}{\partial r^2} + (1/r)\frac{\partial U}{\partial r}).
$$</p>
<p>A friend told me that the equation models the <a href="http://en.wikipedi... | <p>That is the heat equation in polar coordinates with axial symmetry. The (isotropic) heat equation without sources or sinks is</p>
<p>$$
\frac{\partial U}{\partial t} - K\nabla^2U =0.
$$</p>
<p>If you look up the Laplacian operator in cylindrical coordinates, you will find that your expression matches this exactly.... | 1,069 |
differential equations | Solving differential equation in perturbation theory | https://physics.stackexchange.com/questions/774534/solving-differential-equation-in-perturbation-theory | <p>The differential equation of an anharmonic Oscillator with Newtonian friction is
<span class="math-container">$$
\ddot{x}+\varepsilon \dot{x}^2+x=0
.$$</span>
The initial conditions of the System are
<span class="math-container">$$
\begin{align*}
x(0)&=1\\
\dot{x}(0)&=1
.\end{align*}
$$</span>
The System ca... | <p>you want to solve this ODE</p>
<p><span class="math-container">$$\ddot x+x=-\sin^2(t)=-\frac 12\,(1-\cos(2t))$$</span></p>
<p>with the initial conditions <span class="math-container">$~x(0)=0~,\dot x(0)=0~$</span></p>
<p>transfer it to Laplace domain ,obtain the partial fraction and transfer each fraction back to ... | 1,070 |
differential equations | Differential equation with step function | https://physics.stackexchange.com/questions/810499/differential-equation-with-step-function | <p>I want to solve the equations of motion for a system with a unit step function. Are there any methods that can be used to solve these? As a toy model, I picked a sliding mass bouncing off a spring.
The setup for the problem is:
<span class="math-container">$$mx''= -\Theta(x)kx \\ x(0)=u \qquad u > 0 \\ x'(0)=-v ... | 1,071 | |
differential equations | Differential equation for radiation absorption | https://physics.stackexchange.com/questions/673412/differential-equation-for-radiation-absorption | <p>Let the radiation absorbed by a material be given as a function <span class="math-container">$N(x)$</span>, where <span class="math-container">$x$</span> is the material's layer thickness. In a piece with a thickness of <span class="math-container">$dx$</span>, <span class="math-container">$dN$</span> particles are ... | <p>Given that <span class="math-container">$dN$</span> is proportional to <span class="math-container">$N(x)$</span>, we must include the beam loss with it's passage through the absorbing medium by a distance <span class="math-container">$dx$</span> to get the following:</p>
<p><span class="math-container">$$ dN \propt... | 1,072 |
differential equations | What are the differential equations that model a self-propagating gravitational wave in space-time? | https://physics.stackexchange.com/questions/704462/what-are-the-differential-equations-that-model-a-self-propagating-gravitational | <p>Light is a self-propagating wave, but it's very complicated.</p>
<p>Imagine, if you will, a wave in space-time that by assumption was self-propagating like light, except that it was a <a href="https://en.wikipedia.org/wiki/Gravitational_wave" rel="nofollow noreferrer">gravitational wave</a>.</p>
<p>What are the diff... | <p>Wave equation for Gravitational wave(GW) comes from Einstein field equation in general relativity, with linearized approximation. Einstein equation is originally non-linear DE, but we can approximate it become linear. Set metric:</p>
<p><span class="math-container">$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \; |h_{\... | 1,073 |
differential equations | Specific differential equation in RLC circuit | https://physics.stackexchange.com/questions/112713/specific-differential-equation-in-rlc-circuit | <p>I have been studying differential equations in RLC circuits: specifically I am looking at </p>
<p><strong><em>a generator with fixed EMF $=E$,<br>a capacitor $C$, <br>an inductor with inductance $L$ and internal resistance $r$,<br> and a separate resistor $R$</em></strong> </p>
<p>with the elementary cases account... | <p>It is not clear to me why you want to do such a complicated thing. But if you want to follow this way, a slight easier approach is to resolve for the voltage $V_L$. The KVL for your circuit is
$$\tag{1}
E=V_L+(R+r)i+\frac{q}{C}
$$
Now assuming zero initial conditions you have to express $i$ and $q$ in terms of $V_L$... | 1,074 |
differential equations | Confusion about Coulomb Gauge Differential Equations For $\vec{A}$ and $V$ | https://physics.stackexchange.com/questions/307845/confusion-about-coulomb-gauge-differential-equations-for-veca-and-v | <p>I am currently reading Griffiths, 'Introduction to Electrodynamics', 3rd ed, Chapter 10.1.3, the section on Gauge Invariance, and was reached a point of confusion. In particular, the differential equations that arose from choosing the Coulomb gauge $\nabla \cdot \vec{A}=0$:</p>
<p>$$\nabla^2 V=-\frac{1}{\epsilon_0}... | <p>You have done all the work. Now just set $$\vec{A_c} = \vec{A} + \nabla\lambda_C$$ and $$V_c = V - \frac{\partial\lambda_C}{\partial t}\,,$$ and your "New" equations reduce to the Coulomb gauge equations.</p>
| 1,075 |
differential equations | Differential Equations in a Discharging RC Circuit in Parallel | https://physics.stackexchange.com/questions/556163/differential-equations-in-a-discharging-rc-circuit-in-parallel | <p>Please consider the following RC circuit as context:</p>
<p><a href="https://i.sstatic.net/o0gAI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/o0gAI.png" alt="enter image description here"></a>
Assume that the circuit has been connected for a long time. If switch S has been opened at <span class="m... | <p>I believe that your Professor is correct. The equation by bobD is correct but what you are missing is that the charge on capacitor on that time is Q and therefore current flowing through that instant is <span class="math-container">$d(Q(0)-Q)/dt$</span> and that gives your correct equation.
Even if you consider you... | 1,076 |
differential equations | Existence of a solution for geodesic differential equations for a singular metric | https://physics.stackexchange.com/questions/220298/existence-of-a-solution-for-geodesic-differential-equations-for-a-singular-metri | <p>In order to determine the geodesics, one must solve the following set of differential
equations
\begin{align}
\frac{d^2 x^j}{ds^2} + {j\brace h\,\,k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0,
\end{align}
where ${j\brace h\,\,k}$ is the Christoffel symbol of second kind, which is defined as
\begin{align*}
{j\brace h\,\,k}... | 1,077 | |
differential equations | Non-integrable differential equation and non-holonomic contraints | https://physics.stackexchange.com/questions/482949/non-integrable-differential-equation-and-non-holonomic-contraints | <p><a href="https://i.sstatic.net/4hemJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4hemJ.png" alt="enter image description here"></a></p>
<p>From the constraint <span class="math-container">$v=a\dot{\phi}$</span> of a rolling disk over a plane, where <span class="math-container">$a$</span> is the r... | <p>HINT: Rewrite the first constraint as
<span class="math-container">$$
f \left[ dx +(- a \sin \theta) d\phi + (0) d\theta \right] = 0.
$$</span>
where <span class="math-container">$f(x, \theta, \phi)$</span> is some unknown integrating function. We want to know whether this can be written as
<span class="math-conta... | 1,078 |
differential equations | Solution of quadratic stochastic differential equation | https://physics.stackexchange.com/questions/842189/solution-of-quadratic-stochastic-differential-equation | <p>One can write down the solution of a linear stochastic differential equation in Ito convention of the form
<span class="math-container">$$
d\vec{x} = F\vec{x}dt+G\vec{x}dW
$$</span>
where <span class="math-container">$G,F$</span> are constant matrices and <span class="math-container">$dW$</span> is a Wiener process,... | 1,079 | |
differential equations | Integrating Differential equations in General relativity | https://physics.stackexchange.com/questions/763593/integrating-differential-equations-in-general-relativity | <p>Let's say we have an equation of the form</p>
<p><span class="math-container">\begin{equation}
\nabla _\vec{u}u^\mu(\tau)=F^\mu\big[x(\tau)\big]
\end{equation}</span></p>
<p>The operation on the left hand side is the usual covariant derivative along the worldline</p>
<p><span class="math-container">\begin{equation}
... | 1,080 | |
differential equations | Number of differential equations and unknown functions in spherically symmetric black hole solution | https://physics.stackexchange.com/questions/620661/number-of-differential-equations-and-unknown-functions-in-spherically-symmetric | <p>In General Relativity, when we are obtaining the Schwarzchild solution, we get from Einstein's equation three differential equations but only two unknown functions [A(r) and B(r)]:</p>
<p><span class="math-container">$R_{00}=-\frac{A''}{2B}+\frac{A'}{4B}\left(\frac{A'}{A}+\frac{B'}{B}\right)-\frac{A'}{rB}=0,\\
R_{11... | <p>You're correct that there are only two functions, meaning we only need two differential equations. Therefore one must be redundant, which is the case. Showering this can be slightly awkward though.</p>
<p>One trick is to take the derivative of the equations and work with these too. With the Einstein tensor it's a lo... | 1,081 |
differential equations | Drawing the circuit from a differential equation | https://physics.stackexchange.com/questions/77258/drawing-the-circuit-from-a-differential-equation | <p>Can you please help me in modelling a circuit using the differential equation? In the following equation, $u(t)$ is the input voltage and $y(t)$ is the output voltage.</p>
<p>$$y(t)=2u(t)+3\frac{du(t)}{dt}+4\int_0^tu(t)dt.$$</p>
<p>How do I draw a circuit such that the input voltage $u(t)$ and the output voltage $... | <p>I believe you simply need 4 op-amps. Here is what you need for differentiation and integration.</p>
<p><a href="http://en.wikipedia.org/wiki/Operational_amplifier_applications#Integration_and_differentiation" rel="nofollow">http://en.wikipedia.org/wiki/Operational_amplifier_applications#Integration_and_differentiat... | 1,082 |
differential equations | Differential equation from Schwarzschild metric | https://physics.stackexchange.com/questions/837660/differential-equation-from-schwarzschild-metric | <p>I tried to solve an exercise related to Schwarzschild metric, and at some point found next question <a href="https://physics.stackexchange.com/q/620576/">Question</a></p>
<p>I can't figure out how the first line turns out.</p>
<blockquote>
<p>With studying Schwarzschild metric geodesics one can easily come up with t... | <p>It looks like i found an answer in book</p>
<blockquote>
<p>Chandrasekhar S. The Mathematical Theory of Black. Vol. 1. Cambridge: Oxford Univ. Press, 1983. 107 p.</p>
</blockquote>
<p>, but i still appreciate any help for answering some questions (that are well known for physicist) across the proof.</p>
<p>Let us ha... | 1,083 |
differential equations | Book recommendations for Fourier Series, Dirac Delta Function and Differential Equations? | https://physics.stackexchange.com/questions/518442/book-recommendations-for-fourier-series-dirac-delta-function-and-differential-e | <p>I'm a second-year undergrad and currently taking a course in Mathematical Physics which covers the topics of Dirac delta functions, Fourier series, Fourier transforms and Differential equations. They recommended using Boas' "Mathematical Methods in Physical Sciences" book. However, I find the book too "wishy-washy" ... | <p>For a down to earth but rigorous account distributions and delta functions (but not so much differential equations) you can't beat James Lighthill's <em>Introduction to Fourier analysis and generalised functions</em>, Cambridge University Press. ISBN 978-0-521-05556-7.</p>
<p>The book is quite thin, only 70 pages o... | 1,084 |
differential equations | Understanding the Terms in Coupled Springs Differential Equation | https://physics.stackexchange.com/questions/392542/understanding-the-terms-in-coupled-springs-differential-equation | <p>I am teaching differential equations and I got myself totally confused about the physics of a problem.</p>
<p>Consider a coupled spring system in series: there is a mass $m_1$ on a horizontal track which is connected to a wall by a spring (with natural length $L_1$ and spring constant $k_1$). Also attached to the ... | <p>When you apply Newton's laws of motion you are assuming (usually without even thinking about it) that the frame(s) of reference are inertial frames. </p>
<p>In this case you can think of your coordinates $x_1, \, x_2$ and $y_1$ being measured in inertial frames of reference which are all fixed to the Earth. </p>
... | 1,085 |
differential equations | Dimensional analysis in differential equations | https://physics.stackexchange.com/questions/273711/dimensional-analysis-in-differential-equations | <p>I know how to use Buckingham Pi Theorem to, for example derive from the functional equation for a simple pendelum, with the usual methods also described <a href="https://projects.exeter.ac.uk/fluidflow/Courses/FluidDynamics3211-2/DimensionalAnalysis/dimensionalLecturese4.html" rel="nofollow">here</a></p>
<p>$1=fn\l... | <p>You have done more than half the work yourself. It is convenient to define, $\Pi_1\equiv \sqrt{\frac{g}{L}}t$. There is nothing wrong with the way you have defined it, but my definition reduces work in what follows. Rewrite derivative as:</p>
<p>$\frac{d\theta}{dt}=\frac{d\theta}{d\Pi_1}\frac{d\Pi_1}{dt}=\frac{d\th... | 1,086 |
differential equations | A differential equation of Buckling Rod | https://physics.stackexchange.com/questions/40885/a-differential-equation-of-buckling-rod | <p>I tried to solve a differential equation, but unfortunately got stuck at some point. </p>
<p>The problem is to solve the diff. eq. of hard clamped on both ends rod.
And the force compresses the rod at both ends.
the solution(v(x)) is the value of bending I need.</p>
<p>I assuming, that the differential equation ... | <p>First, the solution to your equation is not exactly what you got, but,</p>
<p>$$v(x) = C_1 \cos ax + C_2 \sin ax + C_3x + C_4$$</p>
<p>where $a^2 = \frac{P}{EI_x}$. And then you need to look more carefully at your boundary conditions...</p>
<p>$$v(0) = C_1 + C_4 = 0,\ C_4 = -C_1$$
$$v'(0) = C_2 a + C_3 = 0,\ C_3 ... | 1,087 |
differential equations | Search for differential equation from Green function | https://physics.stackexchange.com/questions/496283/search-for-differential-equation-from-green-function | <p>Let's consider the following: </p>
<blockquote>
<p>We have a Green function <span class="math-container">$G$</span>, and we want to know what linear differential equation is solved by <span class="math-container">$G$</span>. </p>
</blockquote>
<p>How to do this? The question is: If I know <span class="math-conta... | <p>The following is a bit of an inductive approach and it would probably not work for all Green functions. The basic equation that you want to solve is
<span class="math-container">$$ \hat{D} G(\mathbf{x}) = \delta(\mathbf{x}) , $$</span>
where <span class="math-container">$\hat{D}$</span> is the differential operator ... | 1,088 |
differential equations | Dimensionless expression for differential equation | https://physics.stackexchange.com/questions/521952/dimensionless-expression-for-differential-equation | <p>I am working through <em>Nonlinear Dynamics and Chaos</em> by Steven H Strogatz. In chapter 3.5 (overdampened beads on a rotating hoop), a differential equation is converted into a dimensionless form. I am trying to work out which dimensions the initial equations had, and why the converted form is dimensionless.</p>... | <p>Because you take the derivative with respect to <span class="math-container">$\tau$</span>. Since <span class="math-container">$\tau$</span> is dimensionless, the derivative is too.</p>
| 1,089 |
differential equations | Setting up differential equations for two-level Rabi problem | https://physics.stackexchange.com/questions/388677/setting-up-differential-equations-for-two-level-rabi-problem | <p>I try to follow the derivation of Rabi two-level problem but I went into trouble when attempting to set up the equations as many notes have suggested.</p>
<p>Using the book (Laser cooling and trapping by Metcalf and Straten) I am reading. We start by with writing down Schrodinger's equation for a two-level system w... | <p>You probably implicitly making the assumption that the wavelength of this wave is much larger than the size of the atom, so that $kz \ll 1$. Here is why I think that:</p>
<p>I see that you are using the interaction picture, so that $i\hbar \frac{\partial}{\partial t} \lvert \psi \rangle = H'\lvert \psi \rangle$, th... | 1,090 |
differential equations | Should every physical problem formulated as a differential equation have a mathematical solution? | https://physics.stackexchange.com/questions/354051/should-every-physical-problem-formulated-as-a-differential-equation-have-a-mathe | <p>I encountered the following statement in Boyce's <em>Elementary Differential Equations and Boundary Value Problems</em> : </p>
<blockquote>
<p>Not all differential equations have solutions; nor is the question of existence purely mathematical. If a meaningful physical problem is correctly formulated mathematicall... | <p>Maybe there is more context that qualifies this statement, but taken as is, it's completely false. In general, when we talk about existence of solutions to a differential equation, we're talking about existence given a certain set of boundary conditions. It's perfectly possible, in practical real-world problems, tha... | 1,091 |
differential equations | Differential equation for describing a moving disk | https://physics.stackexchange.com/questions/732826/differential-equation-for-describing-a-moving-disk | <p>I'm doing some self-study on physics and came across this problem:</p>
<blockquote>
<p>A disk rolls without slipping across a horizontal plane. The plane of the disk remains vertical, but it is free to rotate about a vertical axis. What generalized coordinates may be used to describe the motion? Write a differential... | <p><a href="https://i.sstatic.net/tERZN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tERZN.png" alt="enter image description here" /></a></p>
<p>the disk can rotate about the z-axis with the angle <span class="math-container">$~\varphi~$</span> and about the y-axis with the angle <span class="math-con... | 1,092 |
differential equations | Frobenius method for fourth-order differential equation | https://physics.stackexchange.com/questions/839455/frobenius-method-for-fourth-order-differential-equation | <p>I am trying to reproduce some results from a paper
<a href="https://iopscience.iop.org/article/10.1209/epl/i1998-00235-7" rel="nofollow noreferrer">https://iopscience.iop.org/article/10.1209/epl/i1998-00235-7</a></p>
<p>The authors solved a 4th order partial differential equation
<span class="math-container">$$\nabl... | 1,093 | |
differential equations | How are the differential forms for Maxwell's Equations used? | https://physics.stackexchange.com/questions/466189/how-are-the-differential-forms-for-maxwells-equations-used | <p>I am currently reading up on Maxwell's Equations (specifically Ampere's Circuital Law- with Maxwell's Addition) for a presentation on differential equations.</p>
<p>I chose the topic ignorant of how the differential form of these equations are used, and I cannot seem to find a digestable use of their differential f... | <p>The integral forms of Maxwell's equations are fairly useless unless you have situations with very high degrees of symmetry and/or fields aligned along co-ordinate axes. e.g. The beloved examples of undergraduate physics everywhere of spherical and cylindrical charge and current distributions.</p>
<p>Once you move a... | 1,094 |
differential equations | Is it possible that classical propagator be used as an integrating factor for solving differential equations? | https://physics.stackexchange.com/questions/816994/is-it-possible-that-classical-propagator-be-used-as-an-integrating-factor-for-so | <p>I have two questions about the picture.</p>
<p>1)
I think classical propagator itself is not function, is just an operator.</p>
<p>And "(operator)(function)" is not that "(operator)X(function)".</p>
<p>So it seems that the product rule can't be applied in differentiation.</p>
<p>Then, is it possi... | <p>I'm answering, assuming <span class="math-container">$L_0$</span> is a scalar.</p>
<h2>Integration in space</h2>
<blockquote>
<p>Then, is it possible to integrate both sides with respect to x ("dx") while solving a differential equation using integrating factor?</p>
</blockquote>
<p>If you perform this ope... | 1,095 |
differential equations | Physical meaning of this boundary value differential equation | https://physics.stackexchange.com/questions/371393/physical-meaning-of-this-boundary-value-differential-equation | <p>(I originally posted this on math stack exchange but was advised to post it here)</p>
<p>I am considering the following boundary value problem:
$$-\frac{\mathrm{d}}{\mathrm{d}x} \left[ a(x) \frac{\mathrm{d}}{\mathrm{d}x}(u(x)) \right] + c(x)u(x) = f(x),$$
where $x \in [0,1]$ and $u(0) = u(1) = 0.$</p>
<p>I search... | <p>I encountered the type of equation you mentioned in lectures I heard on "Methods on the solutions of ordinary & partial differential equations". As example for its usefulness the "heat equation in thermodynamical equilibrium" was cited. In thermodynamical equilibrium the partial time-derivative of temperature "f... | 1,096 |
differential equations | Two parameter differential equation solution | https://physics.stackexchange.com/questions/810734/two-parameter-differential-equation-solution | <p>I am working on the paper titled "Energetic and entropic effects of bath-induced coherences" (<a href="https://arxiv.org/abs/1905.02013" rel="nofollow noreferrer">https://arxiv.org/abs/1905.02013</a>)and there is a two parameter differential equation for calculating the population rates such that:
<span cl... | 1,097 | |
differential equations | Trouble solving partial differential equation with Laplacian squared | https://physics.stackexchange.com/questions/582593/trouble-solving-partial-differential-equation-with-laplacian-squared | <p>I am working in extensions of General Relativity Theory and at the moment of taking the Newtonian limit of this extension theory (essentialy, mathematically speaking, this is just linearizing the field equations obtained via the variational principle, but this is not important) I arrive to the following partial diff... | <p>Take the Fourier transform of each side with
<span class="math-container">$$
h(x) = \int \tilde h(k) e^{-ikx} \frac{d^3k}{(2\pi)^3}
$$</span>
so that
<span class="math-container">$$
\nabla^2 h(x)= \int \left\{-|k^2|\tilde h(k)\right\} e^{-ikx} \frac{d^3k}{(2\pi)^3}, \quad etc.
$$</span></p>
<p>Here <span class="mat... | 1,098 |
differential equations | Amplitude of Oscillation Without Solving Differential Equation | https://physics.stackexchange.com/questions/406265/amplitude-of-oscillation-without-solving-differential-equation | <p>I am currently working on problem in my own research. There seems to be a weak analogy between my problem and motion on a spring. Therefore, I am exploring this question in regards to a mass oscillating on a spring in hopes to gain further insight into my own system in question.</p>
<p>Here is the idea: We can writ... | <p>The answer is yes, because this equation of motion conserves energy. At any time, $E = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2$ is constant because
$$\frac{dE}{dt} = \dot{x}\left(m\ddot{x}+kx\right) = 0$$</p>
<p>This means if we know the initial conditions $x(t=0)$ and $\dot{x}(t=0)$ we know the energy, and we can ... | 1,099 |
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