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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:
$$L(v)=\frac{1}{2}\langle v,v\...
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Find all conserved quantities in a Lagrangian system
Given a Lagrangian system $(M,\mathcal{L})$, the equation of motion is given by the Euler-Lagrange equation. Usually it's hard to solve this equation directly, so one may try to find some conserved ...
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Need help with an Isoperimetric Problem. Am I doing the correct steps? What am I doing wrong?
I have been given a Functional:
$$
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\...
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$-\int_{X} \log(|\det(\nabla \phi_{t})|) d\mu \rightarrow max$ 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} ...
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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,$$L(...
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Noether's theorem and EL equations for $\int_a^b L\left(x(t),u(t),\frac{u_t}{x_t},\frac{1}{x_t}\frac{d}{dt}\left[\frac{u_t}{x_t}\right]\right)x_t\,dt$
I am looking for a faster way to find the appropriate Euler-Lagrange equations for a variational problem of the form
$$\mathcal{L}[x,u] = \int_a^b L\left(x(t),u(t),\frac{u_t}{x_t},\frac{1}{x_t}\frac{d}...
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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} &...
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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
$$ \left\lbrace \begin{array}{r c l c l}
u_t + H(...
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Euler-Lagrange equation minimal surface/graph
Let $u:[a,b]\to \mathbb{R}$ be differentiable, $$\cal{F}(u):=\int_a^b\sqrt{1+u'(t)^2}dt.$$ Find $u$ with $u(a)=u_0, u(b)=u_1$.
Idea: Minimalizing. It is $0=\frac{d}{ds}\cal{F}(u+s\phi)|_{s=0}=\int_a^...
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Solving ${d^2 x \over dt^2}=-\omega^2x +\alpha x^2,$
On solving a Lagrangian, I obtained the Lagrangian equation of motion as
$${d^2 x \over dt^2}=-\omega^2x +\alpha x^2,$$ Where $\omega$ and $\alpha$ are constants and t is the time.
Could anyone ...
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What conditions do I need on a functional such that Euler Lagrange is consistent with boundary condition?
Suppose we have a functional integral between some end points and in order to find the function which optimizes it, I apply the Euler-Lagrange to the integrand. What I receive is a differential ...
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I have a Variation of Calculus, Euler Lagrange, Lagrange Multiplier problem and I don't know how to continue.
So I can't continue with my work. Don't know if I am just not seeing something under my nose or something.
Given formula:
$$I=\int\limits_0^{\pi/2}[(x'_2)^2-(x_2)^2]dt$$
with auxiliary constraints and ...
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What is the most generic way to write a Lagrangian quadratic in velocities?
I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric.
To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ...
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Find curve that minimizes lenght, with integral constraint
I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ...
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Taking the partial derivative of both sides of an equation [duplicate]
I have this function: $G(x, y, z) = G(x, y, g(x,y))$ and the equation $$G = 0$$
I want to reach a specific equation:
$$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}}*\frac{\partial{...
