Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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0
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1answer
64 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
0
votes
1answer
110 views

Brachistochrone problem

I am trying to solve the Brachistochrone problem. First of all I was asking myself whether there is any good reason that the curve $ x $ we are looking for is in $C^2[0,t_\text{end}] $. Then most ...
2
votes
1answer
358 views

variational calculus

$f,g$ are continuous function $[a,b]$.Suppose $$\int_a^b{f(t)h(t)+g(t)h'(t)}dt = 0$$ for every $h$ belonging to $C^1[a,b]$ with $h'(a)=h'(b)=0$. Why it is true that (1)$\int_a^bf(t)dt=0$---->This is ...
1
vote
1answer
48 views

Some ideas about $H=W$

Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ . Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where ...
1
vote
1answer
138 views

How to write the following expression in index notation?

I would like to know how can I write $ ||\vec{a} \times(\nabla \times \vec{a})||^2 $ and $(\vec{a} \cdot (\nabla \times \vec{a}))^2$ in index notation if $\vec{a}=(a_1,a_2,a_3)$ Thank you for ...
3
votes
1answer
156 views

Computing the Euler Lagrange equations

Let $F(u) = \int_0^1(u'')^2+u^2dx $ on $C^2[0,1]$ satisfying $u(0)=a,u(1)=b,u'(0)=c,u'(1)=d$ where $a,b,c,d \in \mathbb{R}$. If $u_*$ is a minimizer, for $\phi \in C^2[0,1],\ \frac{d}{ds}| _{s=0} ...
2
votes
0answers
86 views

Finding a weak minimum

I have been struggling with the following problem I came across in a textbook. I believe that it is necessary to use the Euler-Lagrange Equation. Any help would be greatly appreciated. Let $F$ be the ...
2
votes
1answer
116 views

Show that the arc length integral is continuous in $C^1$

I came across this question and I am not sure how to prove it. Show that the arc length integral is continuous in $C^1$.
4
votes
2answers
182 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
1
vote
0answers
182 views

How to take the limit of some integral?

$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) ...
8
votes
1answer
349 views

Minimizing Lagrangian with two functions

I read this problem where I have to minimize a functional $E[L]$ using calculus of variations, but I'm not sure what is the procedure to follow. The functional is the expected loss: $$E[L] = ...
5
votes
1answer
379 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
4
votes
1answer
253 views

Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm 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 ...
2
votes
1answer
119 views

Representation for function (“null-Lagrangian”)

Let $L(t,x,p) \in C^m([0,1] \times \mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$, $m\geqslant1$ and define for any $u \in C^1([0,1];\mathbb{R}^n)$ $$ \mathcal{L}u = ...
2
votes
2answers
2k views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
11
votes
2answers
371 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
1
vote
0answers
64 views

How can I integrate this?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
2
votes
0answers
31 views

Generic nonlinear Galerkin question

Suppose $X$ is a reflexive complex Banach space, $S:X\rightarrow \mathbb{C}$ is a Gateaux-differentiable function, and consider the variational problem $$ \text{find } u\in X, \quad\text{such that } ...
1
vote
0answers
83 views

Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
1
vote
1answer
79 views

Direct Method: How to show that F is weakly lower semicontinuous?

Let $1<q<\infty$ be given and minimize $F(v):= \int\limits_a^b \lvert v'(x)\rvert^q\, dx$ in the class $K(\alpha,\beta)=\left\{v\in H^{1,q}((a,b),\mathbb{R}^M : v(a)=\alpha, ...
1
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0answers
50 views

How to show that this solution is a minimum.

I am reading this paper from Peter Hess: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differential Equations 6 (1981), no. 8, 951–961. In the last page ...
2
votes
1answer
166 views

How can I solve an Euler-Lagrange equation that satisfies certain conditions

The idea comes from a recreational math problem-- Place two identical coins side by side and roll one along the circumference of another without slipping, how many revolutions will the rolling coin ...
0
votes
0answers
64 views

Bolza problem - coercivity: I do not understand the solution I got

Consider $$ F(u)=\int\limits_0^1 (1-u'(x)^2)^2+u(x)^2\, dx, u\in W^{1,4}(0,1), u(0)=u(1)=0. $$ Show, that $F$ is coercive. To do so use the Young inequality $$ 2ab\leq\varepsilon ...
1
vote
0answers
244 views

How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{equation} \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
1
vote
1answer
224 views

Partial derivative of an integral operator functional

Suppose $f(x) \equiv f_0(x) + \epsilon t(x)$, where $x,\epsilon \in{\mathbb{R}}$. And let $\mathcal{L}[f(x)] \equiv \int_a^b f(x')dx'$. I want to differentiate $\mathcal{L}$ w.r.t. $\epsilon$. So I ...
1
vote
1answer
46 views

derivatives manipulations

How to show this: $f(x,y+en,y'+en')-f(x,y,y')= en(df/dy)+e(dn/dx)(df/dy')+O(e^2)$ y and n are functions of x, e small constant And y is smooth. What identities or properties are used here?
1
vote
0answers
71 views

Proving that this function must be even (II)

Suppose $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. Also let $\mathbf{x}=(x_1,\ldots,x_d)\in\mathbb{R}^d$. I'd like to prove the following: If ...
5
votes
1answer
97 views

Proving that this function must be even

Let $u:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function that is not identically equal to zero. Suppose further that $u$ is an odd function (ie. $u(\mathbf{x})=-u(-\mathbf{x})$). Let ...
0
votes
1answer
301 views

Show that minimum exists (direct method)

Consider $$ F(v):=\int\limits_0^1\lvert v'(x)\rvert^2\, dx, $$ with $$ \left\{v\in H^{1,2}(0,1), v(0)=0=v(1)\right\}. $$ Show that a minimum exists and use the direct method. When I am right ...
1
vote
1answer
100 views

Direct Method : coercive and wlsc - proof

Consider $F\colon M\subseteq X\to [-\infty,\infty], M\neq\emptyset$. Then $\min\limits_{u\in M}F(u)=\alpha$ has a solution, if 1.) $X$ is reflexive. 2.) $F$ is coercive. 3.) $F$ is weak low semi ...
3
votes
1answer
107 views

Direct Method of Variationcalculation

Consider the Bolza problem $$ \inf\left\{F(u)=\int\limits_0^1 ((1-u'^2)^2+u^2)\, dx, u\in W^{1,4}(0,1), u(0)=0=u(1)\right\}. $$ Show that $\inf F(u)=0$, but that it does not exist an $u_0$ with ...
6
votes
1answer
2k views

Constrained variational problems intuition

Problem: minimise $F(x,y,y')$ over $x$, constrained by $G(x,y,y')=0$. $$J_1(x,y,y')=\large \int_{x_0}^{x_1}F(x,y,y')+ \lambda (x) G(x,y,y')dx$$ I understand the Euler-Lagrange equation and Lagrange ...
2
votes
1answer
81 views

Geodesic First Variation

I'm trying to prove that if the first variation of length vanishes then the curve $\gamma$ must be an affinely parameterised geodesic. In the following $T=\dot{\gamma}$. So I've attacked the ...
4
votes
2answers
261 views

Can calculus of varations be formalised with exterior calculus?

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
1
vote
2answers
90 views

Solving $C_1=4y^2+(y')^2+8y$

While working through the exercises in a book on the Calculus of Variations, I've hit a roadblock in trying to solve this differential equation: $C_1=4y^2+(y')^2+8y$ Let me back-up a bit and fill-in ...
1
vote
0answers
66 views

integer transform

Let be $X$ the following finite set: $X=\{0,1,2,\ldots,63\}$. I want to find two function $f$ and $g$ , where $f,g:X \times X \to Z$. We define $x'=f(x,y)$ and $y'=g(x,y)$. We impose the following ...
7
votes
1answer
354 views

existence of a minimizer for functional

My problem is the following: Show that the mapping $u \rightarrow ||\nabla u||^2 + (fu,u)$ has a minimum $u$ in $M:=\{ w \in H^1(\Omega): ||w||=1\}$ . The function $f$ is in $L^\infty$. I dont see ...
3
votes
1answer
570 views

Rayleigh-Ritz method for an extremum problem

I am trying to use the Rayleigh-Ritz method to calculate an approximate solution to the extremum problem with the following functional: $$ L[y]=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy, $$ $D$ is the ...
0
votes
2answers
101 views

Scaling action and one parameter group action

Cosnider the following scaling action: $$ \epsilon . x = e^{2\epsilon}x, \ \ \ \epsilon. y = e^{\epsilon}y. $$ I have to prove that this is a one paraemter group action. What does this mean? I can ...
0
votes
1answer
120 views

Find the extremal on the unit disc

I need help for finding the extremal of: $$J[u]=\int\int_D (u_x^2+u_y^2) dxdy$$ $D$ is the unit disc i.e. $x^2+y^2 \leq 1.$ My boundary condition is $$u(\cos\theta, \sin\theta)=\sin(n\theta), \ \ ...
9
votes
1answer
163 views

Subdifferential boundary conditions: Testing pointwise or with $L^2$ functions

Let $\phi \colon \mathbb R^n \to \mathbb R$ be convex, proper and lower semi-continuous (lsc). Let $M$ be a measurable subset of $\mathbb R^n$. We can define a functional $\Phi \colon L^2(M) \to ...
4
votes
2answers
94 views

Laplace Boundary Problem

Consider a boundary given by vertices $(0,a)$, $(0,0)$ and $(1,0)$ (an 'L' shaped boundary). The problem is to find the equation that passes between the endpoints $(0,a)$ $(1,0)$ of minimum length ...
0
votes
0answers
32 views

Rate of convergence for a many point object ( function or surface ).

I am aware of the definition of a Rate of convergence for iterative methods involving a single point $$ x_{n+1} = \phi(x_n) ~~~;~~\lim_{n \rightarrow \infty} {{ |x_{n+1} -\alpha|} \over{|x_{n} ...
2
votes
1answer
119 views

Null Lagrangians and “Local Degree”

Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function ...
2
votes
1answer
1k views

How is this second form of the Euler-Lagrange equation arrived at?

The Euler Lagrange equation $\frac{\partial F}{\partial q}-\frac{d \frac{\partial F}{\partial \dot{q}}}{d t}=0$ can also be put in the form $\frac{\partial F}{\partial t}-\frac{d (F- ...
1
vote
0answers
216 views

Differentiating with respect to a function using variable transformation

Earlier I asked a question about differentiating $f(x,y)$ with respect to $x-y$. I am working on the solutions trying to use the hints from earlier questions. Is it correct to do the following: ...
4
votes
1answer
180 views

Variation of the fundamental lemma of calculus of variation

Let $$C^1_0[a,b]:=\{f \ C^1[a,b]|f(a)=f(b)=0\}.$$ Providing $C^1_0[a,b]$ is dense in $L^2[a,b]$, I want to prove the following statement: if for $g,h\in L^2[a,b]$, $$\int_a^b g \phi \,dx =\int_a^b h ...
0
votes
0answers
62 views

The optimization problem

Please, help me solve this problem: $$\int_{0}^{T_0}(xy'-yx')dt \to sup,$$ $(\frac{x'}{a})^2+(\frac{y'}{b})^2 \le1,~$ $x(0)=x(T_0),~$ $y(0)=y(T_0)$
3
votes
1answer
286 views

Minimizing an integral with variable endpoints

I am trying to minimize the following functional: $$ J[y]=\int\limits_0^T{\frac{\sqrt{1+y'(x)^2}}{y(x)}dx}, $$ $$ y(0)=1, ~ T-y(T)=1, $$ where $T$ is variable. Using the necessary conditions I've ...
1
vote
0answers
37 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...