  In Part V of this study, we presented an original Lagrangian approach for
computing the dynamic characteristics along stationary rays, by solving the
linear, second-order Jacobi differential equation, considering four sets of
initial conditions as the basic solutions. We then focused on the computation
of the geometric spreading and identification of caustics, where only the two
point-source basic solutions with their corresponding initial conditions are
required. Solutions of the Jacobi equation represent the normal shift vectors
of the paraxial rays and define the geometry of the ray tube with respect to
the stationary central ray. Rather than the Lagrangian approach, the dynamic
characteristics are traditionally computed with the Hamiltonian approach,
formulated normally in terms of two first-order differential equations, where
the solution variables are the paraxial shifts and paraxial slowness changes
along the ray. In this part (Part VI), we compare and relate the two
approaches. We first combine the two first-order Hamiltonian dynamic equations,
eliminating the paraxial variations of the slowness vector. This leads to a
second-order differential equation in terms of the Hamiltonian shift alone,
whose ray-normal counterpart coincides with the normal shift of the
corresponding Lagrangian solution, while the ray-tangent component does not
affect the Jacobian and the geometric spreading. Comparing the proposed
Lagrangian approach to the dynamic ray tracing with the "classical" Hamiltonian
approach, we demonstrate that they are fully compatible for general anisotropy.
We then derive the two-way relationships between the Hamiltonian's and
Lagrangian's Hessians, which are the core computational elements of dynamic ray
theory. Finally, we demonstrate the relationships between these two types of
the Hessians numerically, for a general triclinic medium.
