  We show that there exists an underlying manifold with a conformal metric and
compatible connection form, and a metric type Hamiltonian (which we call the
geometrical picture) that can be put into correspondence with the usual
Hamilton-Lagrange mechanics. The requirement of dynamical equivalence of the
two types of Hamiltonians, that the momenta generated by the two pictures be
equal for all times, is sufficient to determine an expansion of the conformal
factor, defined on the geometrical coordinate representation, in its domain of
analyticity with coefficients to all orders determined by functions of the
potential of the Hamilton-Lagrange picture, defined on the Hamilton-Lagrange
coordinate representation, and its derivatives. Conversely, if the conformal
function is known, the potential of a Hamilton-Lagrange picture can be
determined in a similar way. We show that arbitrary local variations of the
orbits in the Hamilton-Lagrange picture can be generated by variations along
geodesics in the geometrical picture and establish a correspondence which
provides a basis for understanding how the instability in the geometrical
picture is manifested in the instability of the original Hamiltonian motion.
