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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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