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Discrete Hamiltonian Variational Mechanics and Hamel’s Integrators

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A Correction to this article was published on 15 February 2023

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Abstract

Exact variational integrators were exposed in the context of Lagrangian mechanics in Marsden and West (2001). These integrators sample the trajectories of holonomic mechanical systems and are useful for developing practical mechanical integrators. This paper introduces an exact variational integrator for Hamel’s equations, which are interpreted as a noncanonical form of Hamilton’s equations. This exact Hamel integrator is then adopted for a systematic construction of low-order constraint-preserving integrators for nonholonomic mechanical systems.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

We would like to thank the reviewers for helpful remarks and Benliang Wang for assistance with simulations. The research of SG and DS was partially supported by NSFC Grants 12272037, 12232009, and 11872107. The research of DVZ was partially supported by NSF grants DMS-0908995 and DMS-1211454. DVZ would like to acknowledge support and hospitality of the Beijing Institute of Technology, where a part of this work was carried out.

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Authors and Affiliations

  1. Beijing Key Laboratory on MCAACI, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Shan Gao & Donghua Shi

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Dmitry V. Zenkov

Authors
  1. Shan Gao
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  2. Donghua Shi
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  3. Dmitry V. Zenkov
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Correspondence to Donghua Shi.

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Communicated by Arash Yavari.

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The original online version of this article was revised: Figure 1 has been updated.

Induced Variations and Discrete Hamel’s Equations

Induced Variations and Discrete Hamel’s Equations

Using the discrete kinematic equation (3.3), one can derive the induced variations \(\delta \xi _{k+\alpha }\) and the corresponding version of discrete Hamel’s equations as follows.

Taking a variation of (3.3) gives:

$$\begin{aligned} \tfrac{1}{\Delta t}\big (\delta x_{k+1} - \delta x_k\big )&= \delta \Psi _{x_{k+\alpha }}\,\xi _{k+\alpha } + \Psi _{x_{k+\alpha }}\,\delta \xi _{k+\alpha }\\&= {\textbf{i}}_{\delta x_{k+\alpha }} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha } + \Psi _{x_{k+\alpha }}\,\delta \xi _{k+\alpha }\\&= {\textbf{i}}_{(1-\alpha )\delta x_k+\alpha \delta x_{k+1}} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha }, + \Psi _{x_{k+\alpha }}\,\delta \xi _{k+\alpha } \end{aligned}$$

which implies

$$\begin{aligned} \delta \xi _{k+\alpha } = \Psi _{x_{k+\alpha }}^{-1} \tfrac{1}{\Delta t}\big (\delta x_{k+1} - \delta x_k\big ) - \Psi _{x_{k+\alpha }}^{-1} {\textbf{i}}_{(1-\alpha )\delta x_k+\alpha \delta x_{k+1}} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha }. \end{aligned}$$

Next, using \(\delta x_k = \Psi _{x_k}\eta _k\) and taking a variation of the action sum (3.5), we obtain

$$\begin{aligned} \delta a_{\textrm{d}}&= \delta \sum _{k=0}^{N-1} l _{k+\alpha }\\&= \sum _{k=0}^{N-1} D_1 l _{k+\alpha } \big ((1-\alpha )\delta x_k + \alpha \delta x_{k+1}\big ) + D_2 l_ {k+\alpha }\Psi _{x_{k+\alpha }}^{-1}\big ( \tfrac{1}{\Delta t} (\delta x_{k+1}-\delta x_k) \\&\quad - {\textbf{i}}_{(1-\alpha )\delta x_k+\alpha \delta x_{k+1}} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha })\\&= \sum _{k=1}^{N-1} \bigg ( \Big ((1-\alpha ) D_1 l_ {k+\alpha } \Psi _{x_k} + \alpha D_1 l_ {k-1+\alpha } \Psi _{x_{k}}\\&\quad +\tfrac{1}{\Delta t} \big (D_2 l_ {k-1+\alpha } \Psi _{x_{k-1+\alpha }}^{-1} \Psi _{x_k} - D_2 l_{k+\alpha } \Psi _{x_{k+\alpha }}^{-1} \Psi _{x_k}\big )\Big )\eta _k\\&\quad - (1-\alpha )D_2 l_ {k+\alpha } \Psi _{x_{k+\alpha }}^{-1} {\textbf{i}}_{\Psi _{x_k} \eta _k} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha }\\&\quad - \alpha D_2 l_ {k-1+\alpha } \Psi _{x_{k-1+\alpha }}^{-1} {\textbf{i}}_{\Psi _{x_k} \eta _k} d\Psi _{x_{k-1+\alpha }} \xi _{k-1+\alpha }\bigg ), \end{aligned}$$

where

$$\begin{aligned} l_{k+\alpha } = l_{\textrm{d}} (x_{k+\alpha }, \xi _{k+\alpha }). \end{aligned}$$

Thus, \(\delta a_{\textrm{d}}=0\) for each variation with fixed ends if and only if the equations

$$\begin{aligned}&\tfrac{1}{\Delta t} \big (D_2 l_ {k-1+\alpha } \Psi _{x_{k-1+\alpha }}^{-1} \Psi _{x_k} - D_2 l_ {k+\alpha } \Psi _{x_{k+\alpha }}^{-1} \Psi _{x_k}\big ) + (1-\alpha ) D_1 l_ {k+\alpha } \Psi _{x_k} \\&\quad + \alpha D_1 l_ {k-1+\alpha } \Psi _{x_{k}} - (1-\alpha )D_2 l_ {k+\alpha } \Psi _{x_{k+\alpha }}^{-1} {\textbf{i}}_{\Psi _{x_k}} d\Psi _{x_{k+\alpha }} \xi _{k+\alpha } \\&\quad - \alpha D_2 l_ {k-1+\alpha } \Psi _{x_{k-1+\alpha }}^{-1} {\textbf{i}}_{\Psi _{x_k}} d\Psi _{x_{k-1+\alpha }} \xi _{k-1+\alpha } =0 \end{aligned}$$

hold.

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Gao, S., Shi, D. & Zenkov, D.V. Discrete Hamiltonian Variational Mechanics and Hamel’s Integrators. J Nonlinear Sci 33, 26 (2023). https://doi.org/10.1007/s00332-022-09875-w

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