## Fermat's variational principle for anisotropic inhomogeneous media

**Vlastislav Cerveny**
### Summary

In seismology, Fermat's variational principle has mostly been used
in parameteric form. It
is valid for any parameter *u* used to specify the position of points on curves. The relevant
Lagrangian *L*(*x*^{k},dx^{k}/du),
where *x*^{k}, *k*=1,2,3, are general curvilinear
coordinates, is then a homogeneous function of the first
degree in d*x*^{k}/d*u*. It is shown that the Legendre transform cannot be applied to
this Lagrangian to derive the relevant Hamiltonian
*H*(*x*^{k},*p*_{k}) and Hamiltonian ray
equations. The reason is that the Hessian determinant of the transformation vanishes identically
if the Lagrangian is a homogeneous function of the first degree. The Lagrangians must be modified
so that the Hessian determinant is different from zero. Two such modifications are
proposed in this article. In the first modification, the selected parameter *u* along the curves
is chosen to correspond to travel time *tau*, and the modified Lagrangian
*L*^{M}(*x*^{k},d*x*^{k}/d*tau*)
is introduced by the relation
*L*^{M}(*x*^{k},d*x*^{k}/d*tau*)=^{1}/_{2}[*L*(*x*^{k},d*x*^{k}/d*tau*)]^{2}.
The modified Lagrangian *L*^{M}(*x*^{k},d*x*^{k}/d*tau*)
yields the same Euler-Lagrange equations as the standard parameteric Lagrangian
*L*(*x*^{k},d*x*^{k}/d*u*), but
represents a homogeneous function
of the second order in d*x*^{k}/d*tau*
(not of the first order). Consequently, the relevant
Hessian determinant does not vanish identically. In the
second modification, one of the coordinates *x*^{k}, e.g.,
the coordinate *x*^{3}, is chosen to
represent parameter *u*. Here the relevant Lagrangian
*L*^{R}(*x*^{k},d*x*^{1}/d*x*^{3},d*x*^{2}/d*x*^{3})
is referred to as the reduced Lagrangian. Again, the Hessian determinant does
not identically vanish in this case. In both cases, the Legendre transform can be used to
compute the Hamiltonian from the Lagrangian, and vice versa, and the Hamiltonian canonical equations
can be derived from the Euler-Lagrange equations. The relations between modified Hamiltonians and
Lagrangians are discussed in detail. It is shown that the standard form of the Hamiltonian,
derived from the elastodynamic equation and representing the eikonal equation, which has been
broadly used in the seismic ray method, corresponds to the modified Lagrangian
*L*^{M}(*x*^{k},d*x*^{k}/d*tau*),
not to the standard parameteric Lagrangian
*L*(*x*^{k},d*x*^{k}/d*u*).
It is also shown that the relations
*L*^{M}(*x*^{k},d*x*^{k}/d*tau*)=^{1}/_{2}
and *H*^{M}(*x*^{k},*p*_{k})=^{1}/_{2}
are valid along the whole ray and that they represent the
group velocity surface and the slowness surface, respectively. All procedures and derived
equations are valid for general anisotropic inhomogeneous media, and for general curvilinear
coordinates *x*^{i}.
To make certain procedures and equations more transparent and objective, the
simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special
cases.

### Whole paper

The paper is available in
PostScript (1158 kB !)
and GZIPped PostScript (535 kB).

In: Seismic Waves in Complex 3-D Structures, Report 11,
pp. 211-236, Dep. Geophys., Charles Univ., Prague, 2001.

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