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.
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Lagrangian of Rigid Body
In Quantum Mechanics for Mathematicians, by Leon A. Takhtajan page 13 we can find that the configuration space of the Rigid Body is $M=SO(3)$ and its Lagrangian is: $$L(v)=\frac{1}{2}\langle v,v\... differential-geometry classical-mechanics euler-lagrange-equation- 521
Find all conserved quantities in a Lagrangian system
Given a Lagrangian system $(M,\mathcal{L})$, the equation of motion is given by the Euler-Lagrange equation. Usually it's hard to solve this equation directly, so one may try to find some conserved ... classical-mechanics euler-lagrange-equation invariant-theory integrable-systems d-modules- 137
Need help with an Isoperimetric Problem. Am I doing the correct steps? What am I doing wrong?
I have been given a Functional: $$ I=\int^1_0 \left[(x_1')^2+(x'_2)^2-4tx_2-4x_2\right]dt $$ where $x_1=x_1(t)$ and $x_2=x_2(t)$ And auxiliary constraint: $$ \int^1_0 \left[(x'_1)^2-tx_1'-(x'_2)^2\... calculus-of-variations lagrange-multiplier boundary-value-problem constraints euler-lagrange-equation- 45
$-\int_{X} \log(|\det(\nabla \phi_{t})|) d\mu \rightarrow max$ variational problem with differential constraints
I am doing a derivation on some registration problem. One of the substeps of the algorithm is solving the following variational problem \begin{equation} \begin{array}{rrclcl} \displaystyle \max_{\phi} ... lebesgue-measure calculus-of-variations constraints euler-lagrange-equation- 11
I have a problem solving an Isoperimetric Question using Euler-Lagrange
I have been asked to find the extremums for the functional$$\int \limits _0^1(x')^2+t^2\,dt$$subjected to$$\int \limits _0^1x^2\,dt=2,\quad x(0)=0,\quad x(1)=0,$$with $x=x(t)$ and $x'=x'(t)$. Here,$$L(... calculus-of-variations lagrange-multiplier boundary-value-problem constraints euler-lagrange-equation- 45
Noether's theorem and EL equations for $\int_a^b L\left(x(t),u(t),\frac{u_t}{x_t},\frac{1}{x_t}\frac{d}{dt}\left[\frac{u_t}{x_t}\right]\right)x_t\,dt$
I am looking for a faster way to find the appropriate Euler-Lagrange equations for a variational problem of the form $$\mathcal{L}[x,u] = \int_a^b L\left(x(t),u(t),\frac{u_t}{x_t},\frac{1}{x_t}\frac{d}... calculus-of-variations euler-lagrange-equation- 303
converting a differential algebraic system of equation into a lagrange equation
I have searched in quite a few different books now, but I'm still confused about the following: Consider a differential algebraic system of equation of Hessenberg Index-2 form: \begin{align} \dot{x} &... lagrange-multiplier euler-lagrange-equation differential-algebraic-equations- 147
Verification theorem for non autonomous Hamilton-Jacobi PDE
In my course I have a verification theorem for the non autonomous Hamilton-Jacobi PDE that I don't understand. We are interested in the Cauchy problem $$ \left\lbrace \begin{array}{r c l c l} u_t + H(... partial-differential-equations calculus-of-variations euler-lagrange-equation hamilton-jacobi-equation- 1,417
Euler-Lagrange equation minimal surface/graph
Let $u:[a,b]\to \mathbb{R}$ be differentiable, $$\cal{F}(u):=\int_a^b\sqrt{1+u'(t)^2}dt.$$ Find $u$ with $u(a)=u_0, u(b)=u_1$. Idea: Minimalizing. It is $0=\frac{d}{ds}\cal{F}(u+s\phi)|_{s=0}=\int_a^... calculus-of-variations euler-lagrange-equation- 119
Solving ${d^2 x \over dt^2}=-\omega^2x +\alpha x^2,$
On solving a Lagrangian, I obtained the Lagrangian equation of motion as $${d^2 x \over dt^2}=-\omega^2x +\alpha x^2,$$ Where $\omega$ and $\alpha$ are constants and t is the time. Could anyone ... ordinary-differential-equations derivatives partial-differential-equations euler-lagrange-equation- 71
What conditions do I need on a functional such that Euler Lagrange is consistent with boundary condition?
Suppose we have a functional integral between some end points and in order to find the function which optimizes it, I apply the Euler-Lagrange to the integrand. What I receive is a differential ... calculus-of-variations boundary-value-problem euler-lagrange-equation- 7,041
I have a Variation of Calculus, Euler Lagrange, Lagrange Multiplier problem and I don't know how to continue.
So I can't continue with my work. Don't know if I am just not seeing something under my nose or something. Given formula: $$I=\int\limits_0^{\pi/2}[(x'_2)^2-(x_2)^2]dt$$ with auxiliary constraints and ... calculus-of-variations lagrange-multiplier euler-lagrange-equation- 45
What is the most generic way to write a Lagrangian quadratic in velocities?
I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric. To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ... metric-spaces mathematical-physics classical-mechanics connections euler-lagrange-equation- 21
Find curve that minimizes lenght, with integral constraint
I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ... calculus-of-variations boundary-value-problem euler-lagrange-equation- 796
Taking the partial derivative of both sides of an equation [duplicate]
I have this function: $G(x, y, z) = G(x, y, g(x,y))$ and the equation $$G = 0$$ I want to reach a specific equation: $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}}*\frac{\partial{... multivariable-calculus euler-lagrange-equation- 105
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