Asymmetry Amplification in an Anisotropic Model Hamiltonian

We study a Hamiltonian particle moving in an anisotropic central potential that serves as a minimal model for snap-through–type transitions in asymmetric mechanical systems. For the model Hamiltonian, using the Hamilton–Jacobi equation in polar coordinates, we show that the system is fully separable and obtain closed-form quadratures for both the orbital path and the time evolution in terms of elliptic integrals. Within this analytic framework, we define an “optimal trajectory” as the path requiring the smallest total energy to move from the bottom of a potential well to the boundary separating two symmetric half-planes. We explicitly derive the minimal energy needed for such a transition and identify two distinct contributions: the potential difference between the minimum and the relevant saddle channel, and an additional term associated with a radial barrier at the minimum. The integrability of the model system allows a precise definition of the otherwise somewhat blurred concept of “saddle channel”. A key result of the analysis is the identification of a critical anisotropy value at which the topology of the optimal trajectory changes. Below this threshold, the optimal path passes through a finite saddle; above it, the trajectory crosses the basin boundary only asymptotically. We provide explicit analytic expressions for all regimes and identify three distinct types of behaviour (oscillations, heteroclinic motion, and rotations with respect to the angular coordinate), offering a clear and general template for understanding asymmetry amplification and energy-optimal transitions in nonlinear systems.