Generalized Maupertuis’ principle with applications. (English) Zbl 1269.37030
A method to transform a mechanical Lagrangian system into a geodesic system on some energy hypersurface with higher energy is known as Maupertuis’ principle. The author considers some generalized Maupertuis’ principle in the case of Tonelli Hamiltonians. The author proves that the Mañé supercritical potential is equivalent to the corresponding minimal action with respect to an associated Jacobi-Finsler metric. In this way, under weaker conditions, one obtains some applications of the relations between the integrability of the system and the regularity property of Mather’s \(\alpha\)-function.
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
| 37J50 | Action-minimizing orbits and measures (MSC2010) |
| 70H20 | Hamilton-Jacobi equations in mechanics |
| 70H25 | Hamilton’s principle |
| 37N05 | Dynamical systems in classical and celestial mechanics |
Keywords:
\(\alpha\)-functions; generalized Maupertuis’ principle; Jacobi-Finsler metric; Tonelli Hamiltonians; weak KAM solutionsReferences:
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