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Reduction for constrained variational problems and \(\int (\kappa ^ 2/2)ds\). (English) Zbl 0604.58022
The authors study certain functionals whose domain of definition consists of integral curves of an exterior differential system. First, they extend to this general setting the Marsden-Weinstein reduction for Hamiltonian systems [J. Marsden and A. Weinstein, Rep. Math. Phys. 5, 121-130 (1974; Zbl 0327.58005)]. Second, they use this general method to investigate the global behaviour of the solution curve of the Euler- Lagrange equations associated to the functional \[ (1)\quad \Phi (\gamma)=\int_{\gamma}(\kappa^ 2/2)ds \] defined on immersed curves \(\gamma\) in a surface S of constant curvature.
In the flat case \(S=E^ 2\), denote by \([S^ 1,R^ 2]\) the isotopy classes of immersions \(S^ 1\to R^ 2\) of fixed length. The conclusion of this paper shows that the functional (1) has at least one critical value in each component of \([S^ 1,R^ 2]\). In the case when \(S=H^ 2\), the hyperbolic plane, they show that the Euler-Lagrange equations associated to (1) may be represented as a linear flow on a 2-torus \(T^ 2(\mu)\), provided that an ”energy level” \(\mu\cdot \mu\) lies in the interval (-1,0).
Reviewer: S.Izumiya

MSC:
58E30 Variational principles in infinite-dimensional spaces
58A15 Exterior differential systems (Cartan theory)
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