Jacobi Hamiltonian Integrators: construction and applications

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.

💡 Research Summary

The paper introduces a systematic framework for constructing geometric integrators—named Jacobi Hamiltonian Integrators (JHIs)—that preserve the intrinsic structure of Hamiltonian systems defined on Jacobi manifolds. Jacobi geometry unifies Poisson and contact structures, making it a natural setting for both conservative and dissipative dynamics. Traditional symplectic or Poisson integrators cannot be directly applied to Jacobi systems because they either ignore the extra vector field (E) or fail to respect the homogeneity inherent in contact dynamics.

The authors first “Poissonize” a Jacobi manifold ((J,\Lambda,E)) by embedding it into the product (P=J\times\mathbb{R}) equipped with a homogeneous Poisson bivector (\Pi = t^{-1}\Lambda + \partial_t\wedge E). The original Jacobi Hamiltonian (H) lifts to a 1‑homogeneous Hamiltonian (\hat H(x,t)=t,H(x)) on (P). This construction converts the Jacobi flow into a homogeneous Poisson Hamiltonian flow, allowing the use of Poisson geometry tools while preserving the original dynamics up to a simple scaling.

Next, the paper builds a homogeneous symplectic bi‑realization ((\alpha,\beta):U\subset T^{}P\to P). These source and target maps arise from a homogeneous symplectic spray and satisfy the equivariance conditions (h_{z}\circ\alpha=\alpha\circ T^{}h_{z}) and (h_{z}\circ\beta=\beta\circ T^{}h_{z}), where (h_{z}) denotes the (\mathbb{R}^{\times})-action. Within this groupoid‑like structure, Hamiltonian trajectories are represented as Lagrangian bisections (L_{s}\subset T^{}P). The discrete flow is obtained by approximating these bisections.

The core of the numerical scheme relies on the Magnus expansion. For a time‑dependent Hamiltonian (\hat H(t)), the flow operator can be written as (\exp(\operatorname{ad}{\Omega(t)})) with (\Omega(t)) given by a series of nested Poisson brackets. Truncating the series at order (k) yields a (k)‑th order approximation that remains a Poisson (hence Jacobi) map. The authors define recursive generating functions (S{i}) (with (S_{1}=\hat H)) via
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