  Parts V, VI and VII of this study are dedicated to the computation of
paraxial rays and dynamic characteristics along the stationary rays obtained
numerically in Part III. In this part (Part V), we formulate the linear,
second-order, Jacobi dynamic ray tracing equation. In Part VI, we compare the
Lagrangian and Hamiltonian approaches to the dynamic ray tracing in
heterogeneous isotropic and anisotropic media; we demonstrate that the two
approaches are compatible and derive the relationships between the Lagrangian's
and Hamiltonian's Hessian matrices. In Part VII, we apply a similar
finite-element solver, as used for the kinematic ray tracing, to compute the
dynamic characteristics between the source and any point along the ray. The
dynamic characteristics in our study include the relative geometric spreading
and the phase correction factor due to caustics (i.e., the amplitude and the
phase of the Green's function of waves propagating in 3D heterogeneous
anisotropic media). The basic solution of the Jacobi equation is a shift vector
of a paraxial ray in the plane normal to the ray direction at each point along
the central ray. A general paraxial ray is defined by a linear combination of
up to four basic vector solutions, each corresponds to specific initial
conditions related to the ray coordinates at the source. We define the four
basic solutions with two pairs of initial condition sets: point-source and
plane-wave. For the proposed point-source ray coordinates and initial
conditions, we derive the ray Jacobian and relate it to the relative geometric
spreading for general anisotropy. Finally, we introduce a new dynamic
parameter, referred as the normalized geometrical spreading, considered a
measure of complexity of the propagated wave/ray phenomena, and hence, we
propose using a criterion based on this parameter as a qualifying factor
associated with the given ray solution.
