Fermat's variational principle for anisotropic inhomogeneous media

Vlastislav Cerveny


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(xk,dxk/du), where xk, k=1,2,3, are general curvilinear coordinates, is then a homogeneous function of the first degree in dxk/du. It is shown that the Legendre transform cannot be applied to this Lagrangian to derive the relevant Hamiltonian H(xk,pk) 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 LM(xk,dxk/dtau) is introduced by the relation LM(xk,dxk/dtau)=1/2[L(xk,dxk/dtau)]2. The modified Lagrangian LM(xk,dxk/dtau) yields the same Euler-Lagrange equations as the standard parameteric Lagrangian L(xk,dxk/du), but represents a homogeneous function of the second order in dxk/dtau (not of the first order). Consequently, the relevant Hessian determinant does not vanish identically. In the second modification, one of the coordinates xk, e.g., the coordinate x3, is chosen to represent parameter u. Here the relevant Lagrangian LR(xk,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 LM(xk,dxk/dtau), not to the standard parameteric Lagrangian L(xk,dxk/du). It is also shown that the relations LM(xk,dxk/dtau)=1/2 and HM(xk,pk)=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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In: Seismic Waves in Complex 3-D Structures, Report 11, pp. 211-236, Dep. Geophys., Charles Univ., Prague, 2001.
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