Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

Luis C. García-Naranjo; Mats Vermeeren

doi:10.3934/jcd.2021011

Article Contents

2021, Volume 8, Issue 3: 241-271. Doi: 10.3934/jcd.2021011

This issue Previous Article Next Article On computational Poisson geometry II: Numerical methods

Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

Luis C. García-Naranjo^1, and
Mats Vermeeren^2, ,

1.
Departamento de Matemáticas y Mecánica, IIMAS, UNAM, Apdo. Postal 20-126, 01000 Mexico City, Mexico, Current address: Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste 63, 35121 Padova, Italy
2.
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

^* Corresponding author: Mats Vermeeren
^* Corresponding author: Mats Vermeeren

Received: August 31, 2020

Revised: January 31, 2021

Early access: May 2021

Published: July 2021

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Abstract

We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We construct a family of maps, labeled by an arbitrary $ \ell \in \mathbb{N} $ indicating the desired order of accuracy, and prove that our method is structure preserving in the sense that the discrete flow is interpolated to order $ \ell $ by the flow of a continuous system possessing the same structure as the vector field that is being discretized. In particular, our discretization preserves a smooth measure on the phase space to the arbitrary order $ \ell $. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first instance of a measure preserving discretization (to arbitrary order) of measure preserving nonholonomic systems.

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Mathematics Subject Classification: Primary: 37M15; Secondary: 37C40, 37J60, 70G45.

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Figure 1. Evolution of a spherical cloud (black) of $ 5000 $ points with radius $ 0.3 $ at times $ 0 $, $ 2.75 $, $ \ldots $, $ 13.75 $, projected to the $ (x,p_x) $ and $ (y,p_y) $ planes, using a high-accuracy method. The longer the time, the lighter the color

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Figure 2. Evolution of the volume of a spherical cloud of $ 120 $ points, arranged in a $ 600 $-cell, with radius $ 0.01 $ and centered around $ (x,y,p_x,p_y) = (0,0,1,1) $, using our proposed method $ \Phi_h^{(4)} $ with time step $ h = 0.25 $ for the five variational integrators in (24) and a high-accuracy reference solution. The volume is computed with respect to the measure indicated at the top of each graph

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Figure 3. Application of $ \Phi_h^{(4)} $ to the nonholonomic particle in a harmonic potential for each discretization in (24): overview of the numerical values of the different energy functions involved in the algorithm. The initial condition is $ (x,y,p_x),p_y) = (0,0,1,1) $ and the time step $ h = 0.25 $

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Figure 4. Evolution of the volume of a spherical cloud of $ 120 $ points, arranged in a $ 600 $-cell, with radius $ 0.01 $ and centered around $ (x,y,p_x,p_y) = (0,0,1,1) $, using the method $ \Phi_h^{(0)} $ proposed in [23,37,17] with time step $ h = 0.25 $ for the five variational integrators in (24) and a high-accuracy reference solution. The volume is computed with respect to the measure indicated at the top of each graph

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Figure 5. Application of $ \Phi_h^{(4)} $ to the free nonholonomic particle for each discretization in (24). Overview of the numerical values of the different energy functions involved in the algorithm and error norm of the solutions. The initial condition is $ (x,y,p_x,p_y) = (0,0,1,1) $ and the step size $ h = 0.25 $

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Figure 6. Application of $ \Phi_h^{(0)} $ to the free nonholonomic particle for each discretization in (24): overview of the numerical values of the different energy functions involved in the algorithm and error norm of the solutions. The initial condition and the time step coincide with those of Figure 5

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Table 1. Overview of all relevant Hamiltonians

Eqn.	Notation	Name	Equations of motion
Original system
(3)	$ H $	Conformal Hamiltonian	$ {\bf{i}}_{ X} \Omega = \mathcal{N} \mathsf{d} H $
(5)	$ K_E = \mathcal{N}(H-E) $	Altered Hamiltonian	$ {\bf{i}}_{ X} \Omega = \mathsf{d} K_E $ on $ \{ K_E = 0 \} $
Modified system, interpolating numerical solutions
(16)	$ K_{mod} $	Modified altered Hamiltonian	$ {\bf{i}}_{X_{mod}} \Omega = \mathsf{d} K_{mod} $ on $ \{ K_{mod} = 0 \} $
	$ K_{mod}^{(\ell)} $	Truncated modified altered Hamiltonian
(18)	$ \mathcal{E} $	Modified conformal Hamiltonian	$ {\bf{i}}_{X_{mod}} \Omega = \mathcal{N}_{mod} \mathsf{d} \mathcal{E} $
	$ \mathcal{E}^{(\ell)} $	Truncated modified conformal Hamiltonian

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