{'fraction_non_alphanumeric': 0.019286403085824494, 'fraction_numerical': 0.0, 'mean_word_length': 5.447204968944099, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study
the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the
Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures,
the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Henon-Heiles
model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Henon-Heiles model
are obtained. And the numerical results for the Henon-Heiles model show us the instability of the associated geodesic spreads.