On the characterization of infinitesimal symmetries of the relativistic phase space. (English) Zbl 1339.70036

Summary: The phase space of relativistic particle mechanics is defined as the first jet space of motions regarded as time-like one-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally a generalized contact structure on the odd-dimensional phase space. In the paper, infinitesimal symmetries of the phase structures are characterized. More precisely, it is proved that all phase infinitesimal symmetries are special Hamiltonian lifts of distinguished conserved quantities on the phase space. It is proved that generators of infinitesimal symmetries constitute a Lie algebra with respect to a special bracket. A momentum map for groups of symmetries of the geometric structures is provided.


70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
70H45 Constrained dynamics, Dirac’s theory of constraints
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
58A20 Jets in global analysis
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