Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.
8
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347 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 ...
7
votes
0answers
101 views
Equivalence of variational inequalities
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 ...
6
votes
0answers
167 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.
...
4
votes
0answers
40 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
62 views
Optimizing a functional with a differential equation as a constraint
I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense.
We have a parametric ...
4
votes
0answers
88 views
Noether's theorem
I am now reading the book Calculus of Variations written by Jost and I have a problem in the proof of Noether's theorem:
Theorem 1.5.1. Let $F\in C^2([a, b]\times \mathbb R^d \times \mathbb R^d, ...
4
votes
0answers
116 views
Prove that a flat shape minimizes a functional
The following functional arises in an information theoretic problem that I work on currently.
$$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| ...
4
votes
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161 views
Gradient flow of a surface
I found the following definition in a book (S. Osher, R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces", p. 140):
[the context is reconstruction of surfaces from unorganized point sets]
...
3
votes
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72 views
Green's function for third order boundary value problems
How to find the Green's function $G(t,x)$ for the BVP consisting of the equation :
$$u'''(t)=0 , \quad t\in (0,1)$$
and BC :
$$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$
where $\frac12 < ...
3
votes
0answers
48 views
Different functional brachystochrone
Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
3
votes
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48 views
Can calculus of varations be formalised with exterior calculus?
I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior ...
3
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86 views
Nonlinear BVP pde and variational inequality
Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t
$$-\Delta u_a + \frac{m(u_a)}{a} = f$$
where $m(r) = \begin{cases} r ...
3
votes
0answers
65 views
what is the domain of the Lagrangian of a surface embedding?
If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
3
votes
0answers
113 views
Calculus of Variations: Contains an integral of my goal function
OK, using the calculus of variations, I want to find a function $f$ that maximizes:
$$J = \int_0^n L(x,f(x)) \text{d}x$$.
But $L$ has multiple integrals in it (for example, $\displaystyle \int_0^n y ...
2
votes
0answers
70 views
Derivation of Euler-Lagrange equation
Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation.
If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is
$\dfrac{\partial ...
2
votes
0answers
74 views
Gamma Convergence of functionals on Probability measures
Would be grateful if someone could provide a hint or an appropriate reference for the following.
Notation:
$\mathcal{P}(\mathbb{R}^n)$- Space of probability measures on $\mathbb{R}^n$
...
2
votes
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38 views
Finding a weak minimum
I have been struggling with the following problem I came across in a textbook. I believe that it is necessary to use the Euler-Lagrange Equation. Any help would be greatly appreciated.
Let $F$ be the ...
2
votes
0answers
24 views
Generic nonlinear Galerkin question
Suppose $X$ is a reflexive complex Banach space, $S:X\rightarrow \mathbb{C}$ is a Gateaux-differentiable function, and consider the variational problem
$$ \text{find } u\in X, \quad\text{such that } ...
2
votes
0answers
69 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 ...
2
votes
0answers
83 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 ...
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votes
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37 views
Gradient/sub-differential of a minimum of a function
Suppose I have a function $F(x,D) = ||y-Dx||_2^2$, such that $x^{*}(D)= \displaystyle arg \min_{x} F(x,D)$ (that is given $y$ and for a fixed $D$) and subject to some constraint $h(x) <\epsilon$, ...
2
votes
0answers
99 views
Who came up with the Euler-Lagrange equation first?
Could someone explain who came up with the specific equation first?
http://en.wikipedia.org/wiki/Euler-Lagrange
makes it sound like Lagrange got it first, in 1755, then sent it to Euler.
but:
...
2
votes
0answers
51 views
First Weighted Eigenvalue of the Laplacian
Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$:
$$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
2
votes
0answers
59 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 ...
2
votes
0answers
196 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 ...
2
votes
0answers
96 views
Positive rotational symmetric solution for p-Laplacian
I have the the following problem and I just can't get my head around how to solve it.
Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
2
votes
0answers
73 views
Extension of Uncertainty Relations to a specific potential in Schrödinger Equation
Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
1
vote
0answers
58 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
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vote
0answers
16 views
Calculus of Variations statement of a Singular Value Decomposition?
My previous question on SVD gained very little traction, so I thought I'd try a different version that hopefully has an explicit solution. As noted in the linked question, I am taking a function of ...
1
vote
0answers
23 views
Prove a transformation is a variational symmetry for J
The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1)
Let
$$
J(y)=\int_a^b xy'^2\mathrm{d}x.
$$
Show that the transformation
$$
X=x+\epsilon2x\ ...
1
vote
0answers
63 views
Extremal condition calculus of variations
if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L \, dt. $$ Then I get ...
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vote
0answers
72 views
How to take the limit of some integral?
$$ f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty }
\frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[
\tilde{x}D_{1}\left( x,t\right)
...
1
vote
0answers
50 views
How can I integrate this?
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
1
vote
0answers
43 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 ...
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vote
0answers
35 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 ...
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vote
0answers
105 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
0answers
60 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
...
1
vote
0answers
56 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 ...
1
vote
0answers
47 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 ...
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vote
0answers
69 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: ...
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vote
0answers
30 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 ...
1
vote
0answers
41 views
What is the runing time of this algorithm involving length and depth?
I'm hoping that someone can shed some light on this running time.
I have a "tree", for lack of a better description, that has a length $l$ and depth $d$. I want to maximize the tree size, which ...
1
vote
0answers
28 views
variation of a final state due to changes in period (where the period is a parameter)
I have a simple ordinary differential equation
$\frac{dx}{dt}=f(x,t,p,T)$
$x(0) = x_0$, $x(T) = x_T$
where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks!
NOTE: I ...
1
vote
0answers
148 views
Find the extremal to the functional and discuss whether they provide a max/min
I am having a hard time getting my head around Functionals and Calculus of Variations,
My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
1
vote
0answers
138 views
Closed Geodesics as minimisers of action functional
Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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0answers
107 views
Problem of finding strong maxima or minima of a functional
I have got this problem in exam where I have to to check for strong maxima (or minima) or weak maxima(or minima) of the functional given by
$\int_{0}^{1} (1+x)(y^')^2 dx ~~~~~ y(0) = 0, ~~ y(1) = ...
1
vote
0answers
96 views
A “bounded” constraint in a variational problem
Is there any standard approach to solve the following kind of variational problem?
Maximize $F=\int_0 ^1 L(x,y,y')dx $ subject to the constraint $|\int_0 ^1 M(x,y,y')dx| \lt k$ where $y$ ...
1
vote
0answers
141 views
Finding a force function from bodies in equilibrium
(This is an edited version of the original question, since I'm starting a bounty)
I'm trying to find a function $y$ from given data. Reverse optimization, so to speak.
Say we have two ...
1
vote
0answers
74 views
Extremizing an Integral under a cyclic condition
Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that
$$
...
1
vote
0answers
78 views
Question about ellipses and calculus of variations
I don't know much about calculus of variation, but I think it applies to a problem I've come across. If you have a closed loop in a 2 dimensional space defined by some parametric equation r(t), is ...
