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Results tagged with calculus-of-variations
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user 2622
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
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Singularity at left endpoint for variational calculus problem
The answer is yes, if the interval you mean is $(0,\pi/2)$. That is, in any case, the natural domain on which the Euler-Lagrange equation $EL(x)$ is expected to hold. The reason is that it reflects th …
- 19.8k
3
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Conditions ensuring extrema are twice continuously differentiable?
I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagr …
- 19.8k
1
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Functional Minimization: When is this heuristic rigorous?
A useful and fairly complete reference on this and related questions is Morrey Multiple Integrals in the Calculus of Variations.
http://books.google.com/books?id=-QNKm1PBohsC
- 19.8k
5
votes
Accepted
Euler operator as part of a cochain complex
Yes. The next operator in the sequence is called the Helmholtz operator, followed by higher versions thereof. The main keyword is "variational bicomplex" and a standard reference is
I. M. Anderson, “ …
- 19.8k
1
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Variational principle for relativistic gas dynamics
For now I'll just mention that there's a small literature on variational principles for perfect fluids in relativity, though I'm not an expert on it. Here is a reference that discusses some approaches …
- 19.8k
3
votes
Accepted
Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism
I'm not sure that there is a right answer to your question, since what is and what is not a satisfactory generalization of symplectic geometry can be quite subjective and in the end only can judge wha …
- 19.8k
4
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In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
Let $\mathcal{E} \subset J^{2k}E$ be the submanifold (provided that this subset is a submanifold) of all $2k$-jets sitting in the zero-level set of $\rho(E(\mathscr{L}))$, the PDE submanifold. This is …
- 19.8k
0
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How to solve an optimization problem whose optimization variable is a function?
Differentiating the left-hand side of the second condition I get $b^{-1}(f(x) + \int_x^\infty f(t) dt)$, not just $b^{-1} f(x)$, but it is still $\ge 0$. So the second condition can be replaced by its …
- 19.8k
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Integral identity for critical points of the Ginzburg-Landau functional
If I'm not miscalculating, the variational equation for $\varphi$ gives $\delta E_\epsilon/\delta\varphi = \partial^k (\rho^2 \partial_k \varphi) = 0$. Writing your integral identity as a flux through …
- 19.8k
2
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Degenerate second-order Lagrangians
I think you might find interesting §4.B of Anderson's The Variational Bicomplex [1] (Theorems 4.23, 4.29 and Corollary 4.30, to be more precise). However, these results are in a sense converse to your …
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11
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Why the least action principle is always (?) used in this particular form?
In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does no …
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3
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Accepted
Different smooth structures on the infinite jet bundle (for the purposes of calculus of vari...
The following remarks are based on having previously gone through the literature that you've mentioned also for the purposes of figuring out these differences. It has been a while since then, but the …
- 19.8k
5
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Classification of Lagrangians with given Euler-Lagrange equations
In a sense, all the Lagrangians giving the same Euler-Lagrange equations are exhausted by transformations of your type (b), which adds a total derivative/total divergence/boundary term/... Transforma …
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