Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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9
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0answers
437 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
6
votes
0answers
169 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
6
votes
0answers
199 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
5
votes
0answers
589 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
4
votes
0answers
46 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
4
votes
0answers
32 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
4
votes
0answers
45 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
4
votes
0answers
107 views

Levi-Civita Connection for 2-dimensional Riemannian manifold

I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
4
votes
0answers
172 views

Green's function for third order boundary value problems

How to find the Green's function $G(t,x)$ for the BVP consisting of the equation : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < ...
4
votes
0answers
191 views

Calculus of variations

The particular function that extremises a certain functional $J$ among all the functions that render $L$ to another functional $K$ also gives an extremum to the functional $K$ among all the functions ...
4
votes
0answers
127 views

Prove that a flat shape minimizes a functional

The following functional arises in an information theoretic problem that I work on currently. $$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| ...
4
votes
0answers
95 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
4
votes
0answers
373 views

Constraint of a Lagrange multiplier

My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain ...
4
votes
0answers
126 views

Nonlinear BVP pde and variational inequality

Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t $$-\Delta u_a + \frac{m(u_a)}{a} = f$$ where $m(r) = \begin{cases} r ...
3
votes
0answers
23 views

Prove that two functionals with identical differentials differ by a constant.

I am self-studying Calculus of Variations and am struggling to prove results about the variation of a functional that are analogous to results in elementary analysis about differentials/derivatives. ...
3
votes
0answers
60 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and ...
3
votes
0answers
65 views

Complex solution to Euler-Lagrange equation?

I'm currently working on Calculus of Variations and I came across an integral which I had to minimize. The integral I have to minimize is $$\int_0^1(1+y'^2)^2\,dx$$ After getting the Euler-Lagrange ...
3
votes
0answers
52 views

Calculus of Variations with discontinuous Lagrangian

Consider the classical problem of extremizing a functional of the form $$S[x] = \int_a^b L\left(t,x,\dot{x}\right)\ dt.$$ In almost all cases of consideration, the integrand $L$ is considered to be a ...
3
votes
0answers
114 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
3
votes
0answers
60 views

Time-optimal control - Coupled system of equations, control to the origin

I want to find the time-optimal control to the origin $\underline 0$ for the following: $\dot{x}_1=-3x_1 + x_2$ and $\dot{x}_2 = x_1 - 3x_2 + u$, $|u|\leq 1$ How do I go about doing this. I ...
3
votes
0answers
36 views

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Background An elementary problem in the calculus of variations shows that among all curves joining two points $A$, and $B$ in the first quadrant, the one which generates the surface of minimum area ...
3
votes
0answers
227 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
3
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0answers
77 views

Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} ...
3
votes
0answers
115 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
3
votes
0answers
56 views

Maximize an integral

I have the following integral to maximize but I don't know what to do with this $f(b,y)$ in the first integrand: $$ F=\intop_{a}^{b}[f(b,y)+\intop_{a}^{b}f(x,y)dx]dy $$ a and b are constants. Do I ...
3
votes
0answers
117 views

Calculus of variations for implicitly defined functional

I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's ...
3
votes
0answers
74 views

Single Variable Calculus of Variations Question

Problem: Minimize $I(f)$ subject to the constraint $J(f)\leq 0$, where $$I(f)=\int_{x_1}^{x_2}\frac{dx}{f(x)}\tag{$f:[x_1,x_2]\to \mathbb{R}_{\geq 0}$}$$ ...
3
votes
0answers
109 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
3
votes
0answers
140 views

Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...
3
votes
0answers
72 views

Derivative of infimum in variational problem

Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$. In the expression $$\frac{\partial}{\partial \alpha} \inf_{\phi ...
3
votes
0answers
267 views

Transversality condition equation

I'm somewhat baffled: I have a problem in calculus of variations: $$ \int_0^T \!(x-\dot x^2)dt,\qquad x(0)=0,\qquad x(T)=T^2-2. $$ Let $ F(t,x, \dot x) =x-\dot x^2. $ I calculate all the ...
3
votes
0answers
76 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
3
votes
0answers
74 views

what is the domain of the Lagrangian of a surface embedding?

If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
2
votes
0answers
29 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
2
votes
0answers
66 views

Mathematical definition of the Hamiltonian function.

I'm reading this nice text on Calculus of Variations, by Peter Olver. In page $8$, he calls $$J[u] = \int_a^b L(x,u,u')\,{\rm d}x$$ the objective functional, and the integrand $L(x,u,u')$ the ...
2
votes
0answers
25 views

What is the difference between $L$ and $\mathcal{L}$? How does one find the Lagrangian

I'm following a course of Lagrangian and Hamiltonian mechanics, but I'm getting somewhat confused. Could someone explain the difference between $L$ and $\mathcal{L}$? I'm calling both "the Lagrangian ...
2
votes
0answers
37 views

Convergence in the distributional sense (mean field games dynamics)

I am trying to go through the papers by Gueant, Lions and Lasry on Mean field games. One of their examples is the Mexican wave (which happens in football stadia). Straight to the point the Lagrangian ...
2
votes
0answers
33 views

Calculus of variations with double integral, inner integral's upper limit is outer variable of integration

I want to find the function $\gamma(t)$ that makes the following stationary. $S(t) = \int_0^t \gamma(t'') \Big(\beta\int_0^{t''}(1-\gamma(t'))dt' + e_0\Big) dt''$ Constraints: $\beta\gt0$, ...
2
votes
0answers
118 views

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
2
votes
0answers
71 views

Regularity energy minimizing harmonic maps

I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38). This chapter is dealing with the theory ...
2
votes
0answers
61 views

Is there a way to graphically show that a solution is the minimum or stationary solution to a functional?

I'm looking for the functional analogue to the visual representations of function optimization you most commonly see. To illustrate, if we have some function: $$ f(x) = (x-1)^2+1 $$ We can look at ...
2
votes
0answers
49 views

Doubt in the derivation of the field Euler-Lagrange equations

I'm looking at a derivation of the Euler-Lagrange equations in a field setting, and one step in the proof is continually eluding me. Let $\phi(\vec x,t)$ be a field and $\mathscr ...
2
votes
0answers
113 views

Proof of fundamental lemma of calculus of variation.

Suppose $\Omega$ is an open subset of $\mathbb{R}^n$ and let $L^1_\text{Loc}\Omega$ denote all locally integrable functions on $\Omega$ and $C^{\infty}_0\Omega$ for smooth functions whose support lie ...
2
votes
0answers
60 views

Calculus of variations: time of travel between two points

I'm reading Calculus of Variations by Elsgolc. On the page 35 there is example number 7. Let me introduce the problem. We have a functional given by: $v(y(x)) = \int_{x_0}^{x_1}F(x,y,y')dx$ If $F$ ...
2
votes
0answers
52 views

Non-differentiable variational calculus (Dido's problem)

I wonder what is the alternative to Euler-Lagrange equations when we have non-differentiability issues. I'll give an example: Dido's problem can be stated as: Find the figure bounded by a line ...
2
votes
0answers
48 views

green function, functional derivative

I am trying to find ${\delta F}/{\delta u}$ for the functional: $F[u]=\int u(x)\int G(x,y)u(y)dy dx $ G is green function for laplace operator. is there Euler-Lagrange version for double intrgral? ...
2
votes
0answers
71 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
2
votes
0answers
64 views

Calculus of Variation - Question with Integral constraint

I'm stuck on this question, Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$ I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
2
votes
0answers
50 views

Sufficient conditions for the use of the Beltrami identity

For reference, I shall use the notation used in the wikipedia article for the Beltrami identity in the application section (http://en.wikipedia.org/wiki/Beltrami_identity) In the article, the ...
2
votes
0answers
90 views

Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...