A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks
Abstract
Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the 2n projection variables. For constant friction the step rescales ω=dη by the exact factor e−γτ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an O(τ3) one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.
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BibTeX
@misc{loeragaleana2026projected,
title = {{A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks}},
author = {Lorena Loera-Galeana and Santiago Mejía and Espartaco Alvarado and Héctor Medel-Cobaxin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.17198},
eprint = {2608.17198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17198},
}
RIS
TY - UNPB
TI - A Projected Semiexplicit Integrator for Dissipative Systems with Configuration-Dependent Kinetic Energy: Contact-Herglotz Formulation and Benchmarks
AU - Lorena Loera-Galeana
AU - Santiago Mejía
AU - Espartaco Alvarado
AU - Héctor Medel-Cobaxin
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.17198
UR - https://arxiv.org/abs/2608.17198
AB - Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the 2n projection variables. For constant friction the step rescales ω=dη by the exact factor e^−γτ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an O(τ^3) one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.
ER -