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Invariant variational problems on linear frame bundles. (English) Zbl 1029.70013

The authors describe the Hamiltonian structure of actions defined by Lagrangians corresponding to the natural basis of diff \(M\)-invariant Lagrangians on a 1-jet bundle of linear frames of a manifold \(M\). It is shown that there are two types of variational problems. The corresponding field equations are obtained, and it is shown that these are underdetermined nonlinear systems of partial differential equations. Infinitesimal symmetries and Noether invariants of the systems are also studied, and it is shown that the Noether invariant of every vertical symmetry vanishes.

MSC:

70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
58Z05 Applications of global analysis to the sciences
70H30 Other variational principles in mechanics
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