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

Abstract

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.

Acknowledgments

This paper has been inspired by the discussions with Bartolomé Coll and José Maria Pozo in Salamanca in January 2005. I am grateful to Ivan Pšenčík whose suggestions made it possible for me to improve the paper.

The research has been supported by the Grant Agency of the Czech Republic under contracts 205/07/0032 and P210/10/0736, by the Ministry of Education of the Czech Republic within research project MSM0021620860, and by the members of the consortium “Seismic Waves in Complex 3-D Structures” (see http://sw3d.cz).