##
**Relations between constants of motion and conserved functions.**
*(English)*
Zbl 1374.70055

The framework of the paper is a general relativistic spacetime \((M,g)\), consisting of an oriented and time-oriented 4-dimensional Lorentzian manifold. As phase space of particle dynamics, the paper deals with the even-dimensional cotangent bundle \(T^*E\) of the spacetime, or the odd-dimensional 1st jet space \(J_1E\) of motions. They are, respectively, equipped with a symplectic and a contact structure induced by the metric.

It is well known that the infinitesimal symmetries of the kinetic energy function on \(T^*E\) are the Hamiltonian lifts of functions which are constant on geodesic curves. In particular, the infinitesimal symmetries which are polynomial with respect to the fibres of the bundle \(T^*E \to E\) are generated by Killing \(k\)-vector fields, with \(k\geq 1\), (see, for instance, [M. Crampin, Rep. Math. Phys. 20, 31–40 (1984; Zbl 0551.58019)]).

On the other hand, the infinitesimal symmetries of the contact structure of \(J_1E\) are the Hamilton-Jacobi lifts of functions which are conserved by the Reeb vector field. Moreover, these functions are the pullbacks of functions which are constant on geodesics and the infinitesimal symmetries of the contact structure are generated by Killing \(k\)-vector fields. For \(k=1\), the infinitesimal symmetries are projectable, and, for \(k>1\), the infinitesimal symmetries are hidden.

Then, the paper considers the generalised geometric structures on the cotangent bundle \(T^*E\) and on the 1st jet space \(J_1E\) generated by the metric \(g\) and an electromagnetic 2-form \(F\). In this case, the functions which are constant of motion are generated by Killing-Maxwell \(k\)-vector fields, but the pullback of such functions does not generate hidden infinitesimal symmetries of the geometric structure of \(J_1E\).

It is well known that the infinitesimal symmetries of the kinetic energy function on \(T^*E\) are the Hamiltonian lifts of functions which are constant on geodesic curves. In particular, the infinitesimal symmetries which are polynomial with respect to the fibres of the bundle \(T^*E \to E\) are generated by Killing \(k\)-vector fields, with \(k\geq 1\), (see, for instance, [M. Crampin, Rep. Math. Phys. 20, 31–40 (1984; Zbl 0551.58019)]).

On the other hand, the infinitesimal symmetries of the contact structure of \(J_1E\) are the Hamilton-Jacobi lifts of functions which are conserved by the Reeb vector field. Moreover, these functions are the pullbacks of functions which are constant on geodesics and the infinitesimal symmetries of the contact structure are generated by Killing \(k\)-vector fields. For \(k=1\), the infinitesimal symmetries are projectable, and, for \(k>1\), the infinitesimal symmetries are hidden.

Then, the paper considers the generalised geometric structures on the cotangent bundle \(T^*E\) and on the 1st jet space \(J_1E\) generated by the metric \(g\) and an electromagnetic 2-form \(F\). In this case, the functions which are constant of motion are generated by Killing-Maxwell \(k\)-vector fields, but the pullback of such functions does not generate hidden infinitesimal symmetries of the geometric structure of \(J_1E\).

Reviewer: Marco Modugno (Firenze)

### MSC:

70H40 | Relativistic dynamics for problems in Hamiltonian and Lagrangian 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 |