  A new perspective on the classical mechanical formulation of particle
trajectories in lorentz-violating theories is presented. Using the extended
hamiltonian formalism, a Legendre Transformation between the associated
covariant Lagrangian and Hamiltonian varieties is constructed. This approach
enables calculation of trajectories using hamilton's equations in momentum
space and the Euler-Lagrange equations in velocity space away from certain
singular points that arise in the theory. Singular points are naturally
de-singularized by requiring the trajectories to be smooth functions of both
velocity and momentum variables. In addition, it is possible to identify
specific sheets of the dispersion relations that correspond to specific
solutions for the lagrangian. Examples corresponding to bipartite Finsler
functions are computed in detail. A direct connection between the lagrangians
and the field-theoretic solutions to the Dirac equation is also established for
a special case.
