Remarks on structures and preservation in forced discrete mechanical systems of Routh type

TL;DR

The paper develops a geometric framework for forced discrete mechanical systems of Routh type, showing that their time evolution preserves a symplectic form on $Q\times Q$ that arises as a pullback via the forced discrete Legendre transform of a canonical form on $T^*Q$ modified by a magnetic term. It extends symmetry reduction to not-necessarily-abelian groups through an affine discrete connection and discrete momentum maps, deriving a discrete Routh reduction that yields a reduced forced discrete system on $Q/G_\mu$ with a preserved reduced symplectic structure tied to Marsden–Weinstein reduction. Key contributions include explicit expressions for the symplectic forms $\omega_f^\pm$, the reduction by isotropy subgroups $G_\mu$, the identification of the reduced Routh force $\breve{f}_\mu$ as a Routh-type force, and the demonstration that the reduced space carries a symplectic form $\breve{\omega}^+$ compatible with a magnetic term $\beta_\mu$. The numerical experiments with a central potential illustrate the compatibility and limitations of discretization and reduction, showing that a Midpoint discretization preserves the reduced symplectic structure while energy behavior depends on the integrator, and highlighting the potential for structure-preserving algorithms in reduced forced discrete systems. Overall, the work bridges discrete Routh reduction with nonabelian symmetry, clarifies how magnetic-type corrections emerge in the discrete setting, and provides practical routes for geometry-preserving simulations of reduced FDMS.

Abstract

We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete Legendre transform of the canonical symplectic structure on $T^*Q$ modified by a "magnetic term". One example of this type of system is provided by the Lagrangian reduction of a symmetric (unforced) discrete mechanical system in the Routh style. In this particular case, we do not reduce by the full symmetry group but, rather, by an appropriate isotropy subgroup. In this context, the preserved symplectic structure can be alternatively seen as the Marsden-Weinstein reduction of the canonical symplectic structure $ω_{L_d}$ on $Q\times Q$.

Figures (2)

• Figure 1: Error in $r$ in the evolution of the reduced systems computed using RK4 (initial conditions $r_0 := 0.2$ and $\dot{r}_0 := 0.01$) and using MP (initial conditions $r_0 := 0.2$ and $r_1 := 0.201$). In both cases, $h := 0.2$.
• Figure 2: Comparison of the evolution of $\mathfrak{E}^\mu$: exact value (straight blue line), RK4 (brown squares) and MP (small green dots). In all cases, $h := 0.2$.

Theorems & Definitions (89)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Definition 2.9
• Remark 2.10