Fermat's variational principle for anisotropic inhomogeneous media

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 dx^k/du. 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,dx^k/dtau) is introduced by the relation L^M(x^k,dx^k/dtau)=¹/₂[L(x^k,dx^k/dtau)]². The modified Lagrangian L^M(x^k,dx^k/dtau) yields the same Euler-Lagrange equations as the standard parameteric Lagrangian L(x^k,dx^k/du), but represents a homogeneous function of the second order in dx^k/dtau (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 x³, is chosen to represent parameter u. Here the relevant Lagrangian L^R(x^k,dx¹/dx³,dx²/dx³) 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,dx^k/dtau), not to the standard parameteric Lagrangian L(x^k,dx^k/du). It is also shown that the relations L^M(x^k,dx^k/dtau)=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 xⁱ. 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.

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