  In this article I discuss the Jacobi metric in various contexts such as
momentum constraint, Eisenhart lift, frame dragging effect, and Randers-Finsler
metrics that share a common Jacobi metric. First, I introduce the constraint
with relativistic momentum, demonstrate how it can act as a generator of
equations of motion, followed by two applications: the derivation of harmonic
oscillator, and analysis of Schwarzschild Randers-Finsler metric. Then I
present derivation of the Jacobi metric using my constraint formulation,
comparing it for null curves to optical metrics, and discuss a Hamiltonian
mechanics approach to frame dragging, demonstrating how Jacobi metric proves
the constraint more suitable when the Hamiltonian is unavailable. I next
demonstrate the limitations of the Eisenhart lift for Randers-Finsler metrics,
and discuss an alternative via Jacobi metric. Finally, I discuss metrics
sharing a common Jacobi metric considering examples of the
Schwarzschild-Painl\'eve metric and the Kerr metric.
