Hamiltonian formalism

Within the context of the Hamiltonian formalism the travel time can be written as
 (A.18)

where represents the system's Hamiltonian:
 (A.19)

or compactly, in vector notation:
 (A.20)

The perturbation in travel time becomes

where

It follows then that

where

Let us notice now that

so the perturbation in travel time becomes

According to Fermat's principle , which allows to infer the following system of equations
 (A.21)

The Hamiltonian can equally be rewritten in order to proceed with an integration along travel time, and by substituting the components of sound slowness with the components of the wavenumber. In fact, by taking into account that

and that

one can obtain the Hamiltonian[17]A.1:
 (A.22)

(this expression corresponds to the Hamiltonian given by Eq.(A.19), multiplied by the factor ); the system of equations becomes

As shown by the two-dimensional case with cylindrical symmetry follows automatically from this case as

There are also alternativa approaches, which consider a Hamiltonian written in terms of or . For the first case and for two-dimensional case with cylindrical symmetry one can obtain the Hamiltonian

 (A.23)

related to the system of equations

as for the second case the Hamiltonian and associated system of equations correspond to[18]
 (A.24)

and

Orlando Camargo Rodríguez 2012-06-21