  Using the methods of symplectic geometry, we establish the existence of a
canonical transformation from potential model Hamiltonians of standard form in
a Euclidean space to an equivalent geometrical form on a manifold, where the
corresponding motions are along geodesic curves. The advantage of this
representation is that it admits the computation of geometric deviation as a
test for local stability, shown in previous studies to be a very effective
criterion for the stability of the orbits generated by the potential model
Hamiltonian. We describe here an algorithm for finding the generating function
for the canonical transformation and describe some of the properties of this
mapping under local diffeomorphisms. We give a convergence proof for this
algorithm for the one-dimensional case, and provide a precise geometric
formulation of geodesic deviation which relates the stability of the motion in
the geometric form to that of the Hamiltonian standard form. We discuss the
relation of bounded domains in the two representations for which Morse theory
would be applicable. Numerical computations for some interesting examples will
be presented in forthcoming papers.
