## Fermat's variational principle for anisotropic inhomogeneous media

**Vlastislav Cerveny**
### Summary

Fermat's variational principle states that the signal propagates
from point *S* to *R* along a curve which renders Fermat's
functional *I*(*l*) stationary.
Fermat's functional *I*(*l*)
depends on curves *l* which connect points *S* and *R*, and
represents the travel times from *S* to *R* along *l*.
In seismology, it is mostly expressed by the integral
*I*(*l*)=*integral*_{S}^{R}L(*x*^{k},*x*^{k}')d*u*,
taken along curve *l*,
where *L*(*x*^{k},*x*^{k}')
is the relevant Lagrangian, *x*^{k} are
coordinates, *u* is a parameter used to specify the position of
points along *l*, and
*x*^{k}'=d*x*^{k}/d*u*.
If Lagrangian *L*(*x*^{k},*x*^{k}')
is a homogeneous function of the first degree in *x*^{k}',
Fermat's principle is valid for
arbitrary monotonic parameter *u*. We then speak of the first-degree
Lagrangian *L*^{(1)}(*x*^{k},*x*^{k}').
It is shown that the conventional
Legendre transform cannot be applied to the first-degree Lagrangian
*L*^{(1)}(*x*^{k},*x*^{k}')
to derive the relevant Hamiltonian
*H*^{(1)}(*x*^{k},*p*_{k}), and
Hamiltonian ray equations. The reason is that the Hessian determinant
of the transform vanishes identically for first-degree
Lagrangians *L*^{(1)}(*x*^{k},*x*^{k}').
The Lagrangians must be modified so that the Hessian
determinant is different from zero. A modification to overcome this
difficulty is proposed in this article, and is based on second-degree
Lagrangians *L*^{(2)}. Parameter *u* along the curves is
taken to correspond to travel time *tau*, and the second-degree
Lagrangian
*L*^{(2)}(*x*^{k},*x*^{.k})
is then introduced by the relation
*L*^{(2)}(*x*^{k},*x*^{.k})=^{1}/_{2}[*L*^{(1)}(*x*^{k},*x*^{.k})]^{2},
with *x*^{.k}=d*x*^{k}/d*tau*. The
second-degree Lagrangian
*L*^{(2)}(*x*^{k},*x*^{.k})
yields the same Euler-Lagrange equations
for rays as the first-degree Lagrangian
*L*^{(1)}(*x*^{k},*x*^{.k}).
The relevant Hessian determinant, however, does not vanish identically.
Consequently, the
Legendre transform can then be used to compute Hamiltonian
*H*^{(2)}(*x*^{k},*p*_{k})
from Lagrangian
*L*^{(2)}(*x*^{k},*x*^{.k}),
and vice versa, and the Hamiltonian canonical equations
can be derived from the Euler-Lagrange equations.
Both *L*^{(2)}(*x*^{k},*x*^{.k})
and *H*^{(2)}(*x*^{k},*p*_{k})
can be expressed in terms of the wave propagation metric tensor
*g*_{ij}(*x*^{k},*x*^{.k}),
which depends not only on position *x*^{k}, but also on
the direction of vector *x*^{.k}.
It is defined in a Finsler space, in which the distance is measured
by the travel time. 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 second-degree Lagrangian
*L*^{(2)}(*x*^{k},*x*^{.k}),
not to the first-degree Lagrangian
*L*^{(1)}(*x*^{k},*x*^{.k}).
It is also shown that relations
*L*^{(2)}(*x*^{k},*x*^{.k})=^{1}/_{2}
and
*H*^{(2)}(*x*^{k},*p*_{k})=^{1}/_{2}
are valid at any point of the 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.

### Keywords

Fermat's principle, anisotropic media, Lagrangian, Hamiltonian,
Finsler space, wave propagation metric tensor.

### Whole paper

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*Studia Geophysica et Geodaetica*, **46** (2002), 567-588.

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