Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Variational Discretizations for Hamiltonian Systems

Yihan Shen, Yajuan Sun

Abstract

In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the variational principle and the splitting technique, we construct variational integrators and prove their equivalence to the composition of explicit symplectic methods. We apply the newly derived variational integrators to the Kepler problem and demonstrate their effectiveness in numerical simulations. Moreover, using the modified Lagrangian, we analyze the dynamical behavior of the numerical solutions in preserving the Laplace--Runge--Lenz (LRL) vector.

Variational Discretizations for Hamiltonian Systems

Abstract

In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the variational principle and the splitting technique, we construct variational integrators and prove their equivalence to the composition of explicit symplectic methods. We apply the newly derived variational integrators to the Kepler problem and demonstrate their effectiveness in numerical simulations. Moreover, using the modified Lagrangian, we analyze the dynamical behavior of the numerical solutions in preserving the Laplace--Runge--Lenz (LRL) vector.

Paper Structure

This paper contains 6 sections, 19 theorems, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Variational Principle in Lagrangian Formalism
  3. Discrete Lagrangian mechanics
  4. Modified equations for variational integrator
  5. Numerical Experiments
  6. Conclusion

Key Result

Theorem 2.3

Let be a differential system, where $[x]$ contains $t$, $x$, and all the derivatives of $x$. The differential equation $N[x] = 0$ can be derived from a variational principle with the Lagrangian $L$ if the variational derivative of $N$ is self-adjoint, i.e., $N_x = N_x^*$. Furthermore, the Lagrangian $L$

Figures (5)

  • Figure 1: Numerical solutions with time step $h = 0.05$ over 4000 time steps. Red dots represent the exact solution.
  • Figure 2: Numerical error of energy. Top plot: The red line shows the error for the symplectic Euler method, and the blue line shows the error for the first order variational integrator. Bottom plot: The red line shows the error for the Störmer--Verlet method, and the blue line shows the error for the second order variational integrator.
  • Figure 3: Error of eccentricity. Top plot: The red line shows the error for the symplectic Euler method, and the blue line shows the error for the first order variational integrator. Bottom plot: The red line shows the error for the Störmer--Verlet method, and the blue line shows the error for the second order variational integrator.
  • Figure 4: Error of rotation angle. Top plot: The red line shows the error for the symplectic Euler method, and the blue line shows the error for the first order variational integrator. Bottom plot: The red line shows the error for the Störmer--Verlet method, and the blue line shows the error for the second order variational integrator.
  • Figure 5: Convergence rates of the Laplace--Runge--Lenz (LRL) vector over one period of motion. (a) Eccentricity error; (b) Angle error. Solid lines denote reference convergence rates.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8: Kepler's Three Laws
  • ...and 28 more