Equations of motion on the extended Siegel–Jacobi space generated by a linear Hamiltonian

Starting from the case of an extended Siegel–Jacobi upper half-plane, as studied in previous publications [1,2,3], we present a study of the extended Siegel–Jacobi upper half-space of order n, with n>1. Starting from the associated Hamiltonian, which is linear in the generators of the Jacobi group, we write the general formulas for the components of the
total energy function associated with it and then explicitely give their concrete forms. We show that the extended Siegel–Jacobi upper half-space endowed with a certain one-form and a symplectic two-form can be organized as an ACOS manifold, and also as a GTACOS manifold, a fact that helps us formulate the general relations for the equations of motion on it.

Deducing certain correspondences, the concrete expreessions for the equations of motion are computed. The results give a strong generalization of the previously studied case of the equations of motion on the extended Siegel–Jacobi upper half-plane, i.e., when n=1. In conclusion, the main contribution of this work consists of writing the energy functions and deducing the equations of motion on the extended Siegel–Jacobi upper half-space of order n for the coordinates (x,y,q,p,κ), where x,y are symmetric matrices, q,p are column vectors and κ is a real coordinate. The particular case of the extended Siegel–Jacobi upper
half-plane (n=1) is treated separately, showing the correspondences between the formulas in the general case of n>1 and those for n=1. The definitions of ACOS (almost cosymplectic) and GTACOS (generalized transitive almost cosymplectic) manifolds are recalled, as they are essential to our endeavour.

References:
[1]. S. Berceanu, Consequences of the fundamental conjecture for the motion on the Siegel–Jacobi disk, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250076, arXiv:1110.5469 [math.DG].
[2]. S. Berceanu, A convenient coordinatization of Siegel-Jacobi domains, Rev. Math. Phys. 24 (2012) 1250024, arXiv:1204.5610.
[3]. S. Berceanu, Hamiltonian systems on almost cosymplectic manifolds, J. Geom. Phys. 183 (2023) 10470026, arXiv:2201.01962 [math.DG].