Discrete Hamiltonian Variational Mechanics and Hamel’s Integrators

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.

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.