Questions tagged [calculus-of-variations]
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
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Central limit theorem via maximal entropy
Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that:
$$\int_\mathbb{R} \rho(x)\, dx = 1$$
and
$$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$
...
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Largest possible volume of the convex hull of a curve of unit length
What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
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How can you compute the maximum volume of an envelope(used to enclose a letter)?
It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...
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Functional approach vs jet approach to Lagrangian field theory
Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
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Surface equivalent of catenary curve
A catenary curve
is the shape taken by an idealized hanging chain or rope under the influence
of gravity. It has the equation $y= a \cosh (x/a)$.
My question is:
What is the shape taken by an ...
21
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0
answers
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Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
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What was Weierstrass's counterexample to the Dirichlet Principle?
Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
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What are the major differences between real and complex Banach space?
Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
...
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Example of ODE not equivalent to Euler-Lagrange equation
I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation
$$\frac{\partial L}{\...
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answer
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What variational problem does the parabolic suspension bridge solve?
(Posted to MSE here, no answers)
The catenary curve $y(x)$ minimizes the gravitational potential energy
$$\int \rho g y\,ds=\int \rho g y \sqrt{1+y'^2}dx,$$
subject to a fixed length, $L=\int \sqrt{1+...
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answer
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Is a smooth action of a semi-simple Lie group linearizable near a stationary point?
Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
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Smallest area shape that covers all unit length curve
On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this ...
16
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1
answer
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What braking strategy is most fuel-efficient?
You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
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Maxwell equations as Euler-Lagrange equation without electromagnetic potential
In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) ...
15
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Who first resolved Hilbert's 20th problem?
Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lebesgue and Tonelli were pioneers in this area.
In ...
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Good reference for globally formulated calculus of variations on Riemannian manifolds?
I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
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Is $\delta(df \wedge df)=0$ an Euler-Lagrange equation?
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Tweetable way to see Riemannian isometries are harmonic?
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Smooth Riemannian isometries are harmonic. Can one conclude ...
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A riemannian manifold with finitely many closed contractible geodesics
By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if ...
14
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2
answers
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Variation of the Einstein Hilbert action in a coordinate-free way
I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to
provide specific ...
14
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1
answer
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Invariance of the l.h.s. of Euler-Lagrange equation
Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T_xM$. The stationary points of the corresponding integral functional on curves ...
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"Small" maps from sphere to sphere
Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
14
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Area of the minimal surface of a non-planar quadrilateral in 3d
Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
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Beginners text on calculus of variations
I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options.
I work on Machine Learning, and that where I intend to apply this.
...
13
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How to get to the earliest time zone?
You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take?
Formally, fix spherical coordinates $(\theta, \...
13
votes
1
answer
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Choquet theory and Hilbert's fourth problem
The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
13
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Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?
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dual to Hodge theory
Let $(M,g)$ be a closed Riemannian manifold.
In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...
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For what metrics are circles solutions of the isoperimetric problem?
A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
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Generalizing "variation of parameters"
I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$...
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Results about existence/uniqueness of solution to Euler-Lagrange equations?
While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
12
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1
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Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)
Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
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A model of pillows
(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...
11
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answers
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Textbooks or notes on gradient flows in metric spaces
What is a good introduction in gradient flows in metric spaces?
I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...
11
votes
1
answer
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Countable (?) dependent choice
In some circumstances I've been using a form of choice over the first uncountable ordinal knowing a priori that only a countable number of choices were going to be made (without any a priori upper ...
11
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1
answer
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Vector fields on path spaces
I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable ...
11
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1
answer
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What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?
We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$?
The $p$-Laplacian ...
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0
answers
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Maximizing the volume in a family of subsets of a cube
Starting from a question in probability, I arrived to the following optimization problem.
Let $I:=[0, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, $A\subset I^n.$ ...
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Prove/disprove $(\int_{0}^{2 \pi} \!\!\cos f(x) d x)^{2}+(\int_{0}^{2 \pi}\!\!\! \sqrt{(f'(x))^{2}+\sin ^{2} f(x)}dx)^{2}\ge 4\pi^{2}$
This problem has been posted on Math.SE but didn't receive any correct answer after a long time.
Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. ...
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votes
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answers
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Rigorous justification that overdetermined systems do not have a solution
There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
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Good book on Calculus of Variations
What is a good book on the Calculus of Variations, for a second year PhD student?
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Convex curves with many inscribed triangles maximizing perimeter
A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each ...
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1
answer
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Coordinate-free derivation of the Einstein's field equation from the Hilbert action.
It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int_M R \mu_g,$$
is given by ...
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answer
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How to shrink a square with minimal distortion?
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Largest convex hull of a unit length path
What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
9
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1
answer
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Why the least action principle is always (?) used in this particular form?
The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
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4
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Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
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Do these surfaces intersect?
For any real numbers $a_{1},a_{2},\cdots a_{6}$ and $b_{1},b_{2},\cdots b_{6}$
with $\sum_{i=1}^{6}a_{i}^{2}=1$ and $\sum_{i=1}^{6}b_{i}^{2}=1$,
does the equation $$ x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{...
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1
answer
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Which domain maximizes the energy of the Lebesgue measure?
This could be asked in more generality, but let me stick to a concrete case.
Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$...
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Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?
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