# Tagged Questions

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### Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book, Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the page it is good for normal ...
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### Failure of Palais-Smale Condition C and the Mini-Max Principle

To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition ...
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### Is vesica piscis a maximal length curve constrained to two points?

Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a ...
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### Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
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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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### Are all null curves of a Lorentzian metric extrema?

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of $$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$ The so-called "null curves" are ...
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