The Hamiltonian geometry is a generalization of the Finsler geometry, which is in turn a generalization of the Riemann geometry. The Hamiltonian geometry is based on the first-order partial differential Hamilton-Jacobi equations for the characteristic function which represents the distance between two points. The Hamiltonian equations of geodesic deviation may serve to calculate geodesic deviations, amplitudes of waves, and the second-order spatial derivatives of the characteristic function or action. The propagator matrix of geodesic deviation contains all the linearly independent solutions of the linear ordinary differential equations of geodesic deviation.
In this paper, we derive the relations between the propagator matrix of the Hamiltonian equations of geodesic deviation and the second-order spatial derivatives of the characteristic function for a general Hamiltonian function. The derived relations represent the generalization of the analogous relations, previously derived for the Finsler geometry, to an arbitrary Hamiltonian function.
Hamilton-Jacobi equation, geodesics (rays), geodesic deviation, characteristic function, wave propagation, Hamiltonian geometry, Finsler geometry.
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