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Discrete second order constrained Lagrangian systems: first results

Nicolas Borda, Javier Fernandez, Sergio Grillo

Abstract

We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of two mechanical systems: a particle moving in the plane with prescribed signed curvature, and the inertia wheel pendulum with a Lyapunov constraint. In addition, we prove a local existence and uniqueness result for trajectories of DSOCSs. As a first comparison of the underlying geometric structures, we study the symplectic behavior of both SOCSs and DSOCSs.

Discrete second order constrained Lagrangian systems: first results

Abstract

We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of two mechanical systems: a particle moving in the plane with prescribed signed curvature, and the inertia wheel pendulum with a Lyapunov constraint. In addition, we prove a local existence and uniqueness result for trajectories of DSOCSs. As a first comparison of the underlying geometric structures, we study the symplectic behavior of both SOCSs and DSOCSs.

Paper Structure

This paper contains 12 sections, 4 theorems, 36 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Second order constrained Lagrangian systems
  3. Discrete second order constrained Lagrangian systems
  4. Examples
  5. Particle in the plane with prescribed signed curvature
  6. Continuous case
  7. Discrete case
  8. Inertia wheel pendulum with a Lyapunov constraint
  9. Continuous case
  10. Discrete case
  11. Some properties of the discrete flow
  12. Future work

Key Result

Theorem 2.4

Let $(Q,L,C_{K},C_{V})$ be a SOCS with flow $F_{L}:TQ\times\mathbb{R}\rightarrow TQ$ and $t$ be any fixed time. Then, for $\nu\in\Omega^{1}(TQ)$ defined by and where $\gamma$ is the trajectory with initial conditions $(q,\dot{q})$ and, for $s\in[0,t]$,

Figures (6)

  • Figure 1: Scheme of the particle in the plane with prescribed signed curvature. The polar angle $\theta$ of the particle's velocity and the variational constraints at $((x,y),(\dot{x},\dot{y}),(\ddot{x},\ddot{y}))$ are indicated
  • Figure 2: Simulated evolution of the particle using our numerical integrator constructed from a DSOCS for $k=1$, $x(t)=y(t)=0$ and $\dot{x}(0)=\dot {y}(0)=1$. Constant time step used: $h=0.1$. LEFT: trajectory on the plane, RIGHT: comparison between our approximation and the exact solutions of $x$ and $y$ over two time intervals
  • Figure 3: Plot of the $x$-coordinate error vs $h$, using logarithmic scales
  • Figure 4: Scheme of the inertia wheel pendulum. Some of the physical parameters associated to its components (masses, lengths and moments of inertia) as well as the coordinates used are indicated
  • Figure 5: Simulated evolution of $\theta$, $\psi$ and $V$ using our numerical integrator constructed from a DSOCS for the initial conditions $\theta (0)=0.5$, $\psi(0)=0$, $\dot{\theta}(0)=0$, $\dot{\psi}(0)=0.5$. Constant time step used: $h=0.1$. The gray area in the first two graphs corresponds to a fast oscillation
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 2.1: SOCS
  • Definition 2.2: Lagrange--d'Alembert's Principle for SOCSs
  • Remark 2.3
  • Theorem 2.4: Evolution of $\Omega_{L}$
  • proof
  • Remark 2.5
  • Definition 3.1: DSOCS
  • Definition 3.2: Discrete Lagrange--d'Alembert Principle for DSOCSs
  • Theorem 3.3
  • Remark 3.4
  • ...and 8 more