# Tagged Questions

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### Is a smooth action of a semi-simple Lie group linearizable near a staionary 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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### 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 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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### 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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### 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 ...
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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 ...
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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 ...
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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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### Maximizing the volume in a family of subsets of a cube

Starting from a question in probability, one is eventually lead to the following optimization problem. Let $I:=[0,\\, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, ...
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In this recent question Math puzzles for dinner we had a nice time as we were asked to provide new maths puzzles for dinners. I suggested the following: Given three equal sticks, and some ...
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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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### 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 ...
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### Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$? Remark. This question ...
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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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### 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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### Calculating the geodesic equation for a particular set of phase-space coordinates

Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
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### Who came up with the Euler-Lagrange equation?

Which man came up with the solution to the basic Calculus of Variations problem first? http://en.wikipedia.org/wiki/Euler-Lagrange Makes it sound like Lagrange got it first in 1755, then sent it to ...
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### What are the most general types of curves in $\mathbb{R}^2$ for which Gauss-Bonnet holds?

I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has ...
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### Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$
Consider the following equation in $\mathbb{R}^N, N \ge 3$: $$(E) \quad -\Delta u +u=|u|^{p-2}u,$$ where $2 < p < 2^{*} =2N/(N-2)$. Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the ...