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Questions tagged [euler-lagrange-equation]

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. Reference: Wikipedia.

It was developed by Swiss-Russian mathematician Leonhard Euler and French-Italian mathematician Joseph-Louis Lagrange in the 1750s.

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Why is this a first integral? - particle near Schwarzschild black hole

Background I know that the Schwarzschild metric is: $$d s^{2}=c^{2}\left(1-\frac{2 \mu}{r}\right) d t^{2}-\left(1-\frac{2 \mu}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2}$$ I know that if I divide by $d \lambda^2$, I obtain the…
zabop
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Why don't we differentiate velocity wrt position in the Lagrangian?

In Analytic Mechanics, the Lagrangian is taken to be a function of $x$ and $\dot{x}$, where $x$ stands for position and is a function of time and $\dot{x}$ is its derivative wrt time. To set my question, lets consider motion of a particle along a…
mech_love_not_war
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Explanation as to why we treat position and velocity as independent variables in the lagrangian?

Although having studied calculus of variations and lagrangian mechanics, something I've never felt that I've fully justified in my mind is why the lagrangian is a function of position and velocity? My understanding is that the lagrangian…
Will
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Why is $\frac{\operatorname dy'}{\operatorname dy}$ zero, since $y'$ depends on $y$?

I know that $\frac{dy'(x)}{dy}=0$ (where $y'=\frac{dy(x)}{dx}$). The reason explained is that $y'$ does not depend explicitly on $y$. But intuitively, $y'$ depends on $y$, since if you vary $y$ you will modify $y'$. Why is my reasoning wrong (my…
jinawee
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geodesic computation: "energy" minimization versus arc length minimization

Is it true that applying the Euler-Lagrange equation to the integral $E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length integral $L(\gamma)=\int_{t_1}^{t_2}…
J. Heller
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Elliptic regularization of the heat equation

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where $\epsilon > 0$ and $U_T = U \times (0,t]$. Show that…
Cookie
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Is there an invariant way to derive the energy-momentum tensor?

On a (pseudo-)Riemannian manifold $(M, g)$ I can define the following action for any $\phi \in C ^{\infty}(M)$: $$ \mathcal{S}(\phi) = \int_M g(\text{grad }\phi, \text{grad }\phi) \mathrm{d} V. $$ According to the Wikipedia page, the…
Isabella
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Lagrangian Mechanics & Derivatives

I don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it. So for the question: I was doing some review problems on…
Shadow Sniper
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Inequality constraints in calculus of variations

$\def\d{\mathrm{d}}$It turns out that Yuri's answer to my earlier question, whilst correct (and I thank him for his effort), was not quite what I desired. I had not posed the question properly, so I have chosen to re-ask as I am still struggling to…
mch56
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Differential Equation: "I love Physics" - Pendulum meets Capstan equation | Approximation

I came across an amusing YouTube short video (https://www.youtube.com/shorts/Stu0-EiK2Zs) and thought: Let's try to describe this problem physically! My main system is the pendulum with the weight. Here is $l_w$ the length of the pendulum, $R$ the…
Physics
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Can the geodesic equation be used to solve the Brachistochrone Problem?

Assume the initial condition is that a point mass starts at height $y_0$. After descending to height $y < y_0$, we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$. Thus, the displacement element can be written as $$ds^2 = dx^2 + dy^2 = v^2…
Faraz Masroor
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Extremising $\int_0^1 f(x) f(1-x) \ \mathrm{d}x$ subject to length of $f$ and endpoints

I have recently learnt some Calculus of Variations and was trying to apply this to a question I made: Over all functions $f: [0, 1] \to \mathbb{R}$ satisfying $f(0) = f(1) = 0$ with fixed curve length $\ell \geq 1$ (i.e. $\int_0^1 \sqrt{1 +…
Sharky Kesa
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When can the Lagrangian be squared without changing the stationary path?

Assume that the path $y(x)$ makes the functional $$ S[y] = \int _a ^b L(y, y', x) dx$$ stationary. Under what conditions does $$ \int _a ^b L(y, y', x)^2 dx$$ have the same stationary path? And what other functions can be applied to $L$ without…
Luca
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Prove that the minimum of a functional doesn't exist

Prove that there is no smooth solution ho the minimization problem: $$\mathcal{L} (u)= \int_{0}^1 e^{-u'}+u^2 dx$$ Where the admissible space is $X =\{ u \in \mathcal{c}^2 [0,1] | u(0)=0, u(1)=1 \} $ UPDATE: GOAL: I am trying to define a…
Véronique
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Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation (I'm guessing they assume the reader knows it…
Arturo don Juan
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