## Relation between the propagator matrix
of geodesic deviation and the second-order
derivatives of the characteristic function
for a general Hamiltonian function

**Ludek Klimes**
### Summary

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.

### Keywords

Hamilton-Jacobi equation, geodesics (rays), geodesic deviation,
characteristic function, wave propagation, Hamiltonian geometry,
Finsler geometry.

### Whole paper

The paper is available in
PDF (128 kB).

In: Seismic Waves in Complex 3-D Structures, Report 23,
pp. 121-134, Dep. Geophys., Charles Univ., Prague, 2013.