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. The definition of the propagator matrix of geodesic deviation depends on the independent parameter along geodesics.
The previously derived relations between the propagator matrix of the Hamiltonian equations of geodesic deviation and the second-order spatial derivatives of the characteristic function contain the spatial gradients of the independent parameter along the geodesic calculated between two points. In this paper, we derive the equations for calculating the spatial gradients of the independent parameter along geodesics from the propagator matrix of geodesic deviation. These new equations enable us to derive the explicit expressions for the second-order spatial derivatives of the characteristic function in terms of the propagator matrix of geodesic deviation. All equations are derived for a general Hamiltonian function.
Hamilton-Jacobi equation, geodesics (rays), geodesic deviation, characteristic function, wave propagation, Hamiltonian geometry, Finsler geometry.
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