A Geometry of Hamiltonian Mechanics
Year: 2026
Type: article
Abstract: We develop a local, patchwise geometric framework that embeds a broad class of potential Hamiltonian dynamical systems into a family of Riemannian Hamilton patches built over an underlying Gutzwiller manifold. We adopt a conformal (Jacobi) ansatz and a frame-adapted reconstruction procedure, through which we construct, on each patch, a pulled-back metric, along with a reduced (truncated) connection (not a metric-compatible connection) and a corresponding dynamical curvature tensor governing geodesic deviation in the Hamilton coordinates. Then, using the Poisson–Hodge reconstruction, we reconstruct coordinate potentials, enforcing harmonic obstructions, and along with exactness and Jacobian nondegeneracy conditions, we obtain explicit elliptic bounds that control the connection and curvature residuals. On the basis of this construction, we formalize the notion of a Hamilton manifold such that reparametrized geodesics approximate Newton trajectories with controlled acceleration and tolerances. As a generalized structural framework, to promote the local Jacobi reconstructions to a coherent dynamical evolution and provide a dynamical closure, we introduce a patchwise hyperbolic geometric flow for the pullback metric coupled to a kinetic (Vlasov) closure that controls reconstruction and curvature residuals. Under natural regularity, ellipticity, and overlap-tolerance assumptions, together with precise estimates that control the reconstruction and curvature errors, we establish short-time well-posedness of the coupled Vlasov–hyperbolic geometric flow that defines the patchwise Hamilton manifold. Motivated by this construction of the Hamilton manifold with atlas-dependent time, we propose convergence and stability conjectures for dissipative and conservative (non-dissipative) hyperbolic geometric flows. On a single patch, these conjectures characterize local orbital stability (in the sense of coercivity modulo symmetry) and identify local linear instability when unstable linear modes are present. On a finite atlas (the Hamilton manifold with atlas-dependent time), we state conjectures under which local stability propagates to global stability, provided that overlap residuals remain uniformly sufficiently small. The framework identifies the geometric origin of local instability diagnostics used in Hamiltonian mechanics and outlines a practical strategy for verifying stability or instability, numerically or analytically, on finite coverings of configuration space (the Hamilton manifold).
Source: Entropy
DOI: 10.3390/e28040379
Authors: Gil Elgressy, Lawrence Horwitz
Institutions: Bar-Ilan University, Tel Aviv University, Ariel University
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7141492666