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

**Ludek Klimes**
### Summary

In the Finsler geometry, which is a generalization of the Riemann geometry,
the metric tensor also depends on the direction of propagation.
The basics of the Finsler geometry were formulated by
William Rowan Hamilton in 1832.
Hamilton's formulation is based
on the first-order partial differential
Hamilton-Jacobi equations for the characteristic function
which represents the distance between two points.
The characteristic function and geodesics together
with the geodesic deviation
in the Finsler space can be calculated efficiently by Hamilton's method.
The Hamiltonian equations of geodesic deviation
are considerably simpler than
the Riemannian or Finslerian equations of geodesic deviation.
The linear ordinary differential 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 equations of geodesic deviation.

In this paper, we use the Hamiltonian formulation
to derive the relation between
the propagator matrix of geodesic deviation
and the second-order spatial derivatives of the characteristic function
in the Finsler geometry.
We assume that the Hamiltonian function
is a positively homogeneous function of the second degree
with respect to the spatial gradient of the characteristic function,
which corresponds to the Riemannian or Finslerian equations
of geodesics and of geodesic deviation.
The derived equations, which represent the main result of this paper,
are applicable to the Finsler geometry, the Riemann geometry,
and their various applications such as general relativity
or the high-frequency approximations of wave propagation.

### Whole paper

The reprint is available in
PDF (125 kB).

*J. electromagn. Waves Appl.*, **27** (2013), 1589-1601.