A new approach to integrals of discretizations by polarization
Yuri B. Suris
TL;DR
This work addresses the derivation of integrals of motion for polarization-based discretizations of Hamiltonian systems, focusing on the Kahan method for quadratic vector fields. The authors introduce an algebraic framework that decomposes a Hamiltonian $H(x)=H_3(x)+H_2(x)+H_1(x)$ and uses polarization operators $pol_2$ and $pol_3$ to construct discrete invariants. For cubic first-order Hamiltonians, a conserved quantity $H_ extepsilon(x_n,x_{n+1})$ is derived with the explicit form $H_ extepsilon(x_n,x_{n+1}) = rac{1}{ extepsilon} x_n^T J^{-1} x_{n+1} + pol_2 H_2(x_n,x_{n+1}) + 2\,pol_2 H_1(x_n,x_{n+1})$, reducing to $ extepsilon H_ extepsilon(x_n,x_{n+1}) = x_n^T J^{-1} x_{n+1}$ when $H$ is homogeneous of degree 3. For quartic second-order systems, a corresponding invariant $H_ extepsilon(x_{n-1},x_n,x_{n+1})$ is obtained, with a simple homogeneous-degree-4 case giving $ extepsilon^2 H_ extepsilon = x_{n-1}^T K^{-1} x_n - 2 x_{n-1}^T K^{-1} x_{n+1} + x_n^T K^{-1} x_{n+1}$ and links to the continuous integral $4H(x, abla x)$. Overall, the work clarifies the algebraic structure behind discrete integrals of motion from polarization discretizations and suggests avenues for broader applicability and theoretical understanding of discrete geometric integrators.
Abstract
Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.