## Calculation of the spatial gradient
of the independent parameter along geodesics
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.
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.

### Keywords

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

### Whole paper

The paper is available in
PDF (92 kB).

In: Seismic Waves in Complex 3-D Structures, Report 23,
pp. 135-143, Dep. Geophys., Charles Univ., Prague, 2013.