# 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.

864 questions
Filter by
Sorted by
Tagged with
1 vote
29 views

• 367
48 views

### Convex Lagrangian prevents oscillations - why?

We are given an interval $I \subset \mathbb{R},$ the space of $q-$absolutely continuous functions $AC^q(I,\mathbb{R}^{n})$ which satisfy continuity of the function and the existence of its $q-$...
• 103
76 views

### How should I show the following stationary path with a natural boundary condition at $x=0$?

Let $a>0, b>0, c>0$ be constants. Show that the stationary path of the functional $$S[y]=\int_{0}^{a}(y'^2+2by)dx, \quad y(a)=0,$$ with a natural boundary condition at $x=0$ and subject to ...
• 71
1 vote
34 views

• 71
17 views

### Deriving correct integral constraint equation for calculus of variations problem

I have the following calculus of variations problem: $$\mathcal{L}=-2X'(t)\ln{x'(t)}-2Y'(t)\ln{y'(t)}$$ where $x(t)$ and $y(t)$ are the functions I'm interested in, and $X(t)$ and $Y(t)$ are given as ...
• 101
48 views

• 125
41 views

• 135
20 views

### Euler Lagrange Equation on special brachistochrone

I have to use the Euler Lagrange Equation on a special form of the brachistochrone, which includes gravity. So the formular would be: $$T[y]=\int_{a}^{b}\tfrac{\sqrt{1+y'((x))^2}}{\sqrt{(y(x)g(x)}}$$...
126 views

### How should I derive the Euler-Lagrange equation by using the Gateaux differential?

Find the Gateaux differential for the functional $S[y]=y(0)+\frac{1}{2}\int_{0}^{1}(4y'^2+x^2y^2)dx, y(1)=2$ and use it to derive the Euler-Lagrange equation. Be sure to specify all boundary ...
• 71
79 views

### Equivalent Formulations of Variational Problems

This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or ...
• 2,256
1 vote
147 views

• 121
### Minimizer over set of $C^1$ functions
Exercise (1.4) of Renardy and Rogers: "An introduction to partial differential equations" asks to Show that there is an infinite family of minimizers of  J(u) = \int_0^1 (1-u'(t)^2)^2\,...