On some aspects of the discretization of the suslov problem

Figure 1.  In this figure we display the performance of the midpoint rule ($\overline{\mbox{DSP }}(\omega^k,\lambda_{k+1};\omega^{k+1}) = 0$, with inertia matrix $\mathbb{I}$ and initial values $\omega_1(0)$ and $\omega_2(0)$ introduced above) for the nonholonomic rigid body with a time step of size $\epsilon = 10^{-3}$. The solid red line is obtained through a RK4 integrator (which we consider an accurate approximation of the continuous nonlinear dynamics over a short time interval), while the blue dots represent the performance of the midpoint rule. The plots $(a)$ and $(b)$ correspond to the dynamical variables $\omega_1$, $\omega_2$, while $(c)$ displays the Lagrange multipliers $\lambda.$ On the other hand $(d)$ shows the inconsistent multipliers generated by the nonholonomic variational integrator. Finally, $(e)$ and $(f)$ show the preservation of the constraints and the energy $E_l(\hat\omega)$ up through round off errors, respectively.

Figure 2.  This figure displays the comparison between the midpoint rule (the same as in Figure 1) and the variational integrator (37), (38), for a time step of size $\epsilon = 10^0 = 1$ (we recall that this integrator is also order 2 consistent in the dynamical variables). The former is represented by the green points and the latter by the blue ones, while the solid red line still represents the performance of a RK4 method. Variables $\omega_1$ $(a)$, $\omega_2$ $(b)$, $\lambda$ $(c)$ and $E_l$ $(d)$ are displayed, while $(e)$ shows the preservation of the constraints by the variational integrator up through round off errors. We observe a better performance of the variational integrator, mainly with respect to the preservation of energy, a fact which, considering bigger time steps, leads to the conclusion that its convergence to the actual solution is much faster and its long-term behavior is much more accurate.