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

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10
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466 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
120 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 ...
6
votes
0answers
180 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): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
5
votes
0answers
89 views

Does this inequality hold? Proof / Counterexample

Does the following inequality $ \int_0 ^\infty x^2 |\frac{d}{dx}f(x)|^2 dx - \int_0 ^\infty x |f(x)|^2 dx + 2\pi (\int_0 ^\infty x^2 |f(x)|^2 dx) (\int_0 ^\infty x |f(x)|^2 dx) > -\frac{1}{8\pi} ...
5
votes
0answers
655 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
50 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
54 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
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0answers
121 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
199 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
128 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
102 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
443 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
151 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
69 views

Green's first identity and the calculus of variations

UPDATE: I was able to solve this problem using iterative integration by parts. However, I still cannot find how Green's first identity would apply here. Suppose I had a multiple integral over ...
3
votes
0answers
51 views

Example of an oscillation Young measure

I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get ...
3
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41 views

Weakening the Fundamental Lemma of Calculus of Variations

The Fundamental Lemma of Calculus of Variation says that if a continuous function $f$ on an open interval $(a,b)$ satisfies the equality $$\int_{a}^{b} f(x) h(x) = 0$$ for all compactly supported ...
3
votes
0answers
27 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
3
votes
0answers
56 views

Fundamental Lemma of the Calculus of Variations with higher derivatives

The fundamental lemma of the calculus of variations is often presented as: If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...
3
votes
0answers
75 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)= ...
3
votes
0answers
34 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
69 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
165 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
88 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
116 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
46 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
279 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
votes
0answers
146 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
142 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
84 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
62 views

What methods are available for this optimization problem?

I have an intermediate knowledge of the calculus of variations: I can handle constraints in functional or integral forms and extrapolate to multiple variables and functions. If I dig in my notebooks I ...
3
votes
0answers
128 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
181 views

Lagrange multipliers in the context of the calculus of variations

Suppose we wanted to extremise the function (of a finite number of variables) $f$ subject to the constraint $g = 0$. The Lagrange multiplier approach is to extremise without constraint the function ...
3
votes
0answers
78 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
209 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
105 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
491 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
77 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
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0answers
205 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 ...
3
votes
0answers
171 views

Is this calculus of variations intuition justifiable?

I'll preface this by saying that I haven't taken an in depth study of the calculus of variations and have only come across it recently in applications; in depth study is on my to do list. I'm ...
3
votes
0answers
79 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
31 views

Where does Jacobi's accessory equation come from?

I'm reading Charles Fox's An Introduction to the Calculus of Variations and in section 2.4 he just suddenly introduces Jacobi's accessory equation and I don't understand where it's coming from. ...
2
votes
0answers
21 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
2
votes
0answers
39 views

Fundamental lemma of calculus of variation, about hypothesis

We can find on the web several forms of the fundamental lemma of calculus of variation, the simplest one I could find (French wikipedia ) is: for $f\in C^1([a, b])$ $$ \int_a^b f(x) g(x) dx = 0, ...
2
votes
0answers
19 views

Does this mean a new boundary condition for ODE and how to handle it?

In order to to calculate desired light path in continuous medium with gradient refraction index, for schematic see the Figure below. $O:(0,0)$ is the disk center of light source $\odot{O}$ with ...
2
votes
0answers
47 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} ...
2
votes
0answers
37 views

Functional integration and Feynman path integrals in wolfram alpha

Is it possible to do Feynman path integrals in wolfram alpha? Say for a free quantum mechanical particle. The reason I am interested in this is because I would like to see how it arrives at the ...
2
votes
0answers
31 views

Sturm-Liouville Variational Problem

I'm entirely clueless with this problem. No formal training in variational methods. Show that for function $\phi\left ( x \right )$ with $$\phi\left ( a \right )=\phi\left ( b \right )=0$$ and ...
2
votes
0answers
53 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = ...
2
votes
0answers
130 views

Geodesic on the surface of a cone (calculus of variations)

I have been trying to solve an exercice I found on a book. It is about a geodesic on the surface of a cone. The answer is plainly provided at the end of the book without any hint or detail and my ...
2
votes
0answers
26 views

Time-dependent inequalities in optimal controller

I need to build the optimal controller, i.e. one that maximizes: $J = \int_{0}^{t_f} f(u) \mathrm{d}t$ For the following time-dependent system: $\dot{x} = g(x, u, t)$, $x(t) \geq l(t)\; \forall t ...