Articles of calculus of variations

Derivative with respect to $y'$ in the Euler-Lagrange differential equation

I am having trouble understanding the $ \dfrac{\partial L}{\partial y’} $ part in Euler-Lagrange Equation. For example, if $ L = y^2(z) $, what is the symbolic expression for $ \displaystyle\frac{\partial (y^2(z))}{\partial (\frac{\partial y}{\partial z})} $?

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y’)^2 – y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ < $\pi$ weak minimum if ${\alpha}$ > $\pi$ strong minimum if ${\alpha}$ > $\pi$ Comparing the given functional to the standard form $J = \int_{a}^{b} F(x, y, y^{‘})$, we […]

How to check if a function is minimum to functional?

Given $\int_0^1(y’)^3dx$ functional and $y(0) = 0 ,y(1)=1$ conditions. Using Euler–Lagrange equation I have got $y(x)=x$. So $y$ is a stationary point of the functional. How to check if it is the minimum for $y \in C^2[0,1]$ ?

A variation of fundamental lemma of variation of calculus .

I have a question on a variation of the fundamental lemma . If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $ then is it possible that there exist some constant $c$ such that $f(x)=c$ for all $x\in \Omega$ I tried to use mollification on one of the function and […]

Function extremal – calculus of variations

Find a curve passing through (1,2) and (2,4) that is an extremal of the function: $J(x,y’)=\int_1^2 xy'(x)+(y'(x))^2dx$ I don’t know what methods to use at all.

Step in derivation of Euler-Lagrange equations of motion

From Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we have $${\frac{\partial x}{\partial X}=\frac{\partial{\dot x}}{\partial{\dot X}}}$$ (This is just after equation (5) on the page.) I’m having trouble making sense of this. If each of these variables is […]

Calculus of variations – when $y'$ doesn't exist; example: the isoperimetric problem

The isoperimetric problem is the following: Among all curves of length L in the upper half-plane passing through the points $( – a, 0)$ and $(a, 0)$, find the one which together with the interval $[ – a, a]$ encloses the largest area. Solution from Formin and Gelfand’s book of calculus of variations (page 49): […]

What is the solution to the Dido isoperimetric problem when the length is longer than the half-circle?

Given $L$-the length of a curve (single-valued function) passing trough the points $x_1$ and $x_2$ on the $x$-axis. What is the curve $y(x)$ maximizing the area between this curve and the $x$-axis? The solution is, of course, well known: one formulates a variational problem with a constraint $ F[y,y’]=\int_{x_1}^{x_2}\left(y+\lambda\sqrt{1+y’^2(x)}\right)dx, $ which yields an equation of […]

Isoperimetric problem in the calculus of variations

I’m trying to solve the following isoperimetric problem: A plane curve has length $l$ and end points at $(0, 0)$ and $(a, 0)$ on the positive $x$ axis. Show that the area $A$ under this curve is given by $$A = \int_0^l y\sqrt{1 – y'^2}ds,$$ where $y' = dy/ds$. Find the function $y(s)$ and the […]

Differentiating a function of a variable with respect to the variable's derivative

Suppose $x:\mathbb{R}\to \mathbb{R}$ is parameterised by $\lambda$. What does it mean to take a derivative of a function $f(x)$ with respect to $\dot{x} = \frac{dx}{d\lambda}$. i.e. what does $\frac{df(x)}{d\dot{x}}$ mean? How do we compute it? Is $\frac{d}{d\dot{x}}=\frac{d}{d\frac{dx}{d\lambda}}=^{??} \frac{d\lambda}{dx}=^{??} 0$ ??? For example, how would one compute $\frac{d}{d\dot{x}} e^x$? (This question has arisen from an undergraduate […]