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Arbitrarily high-order energy-conserving methods for Hamiltonian problems with quadratic holonomic constraints

P. Amodio, L. Brugnano, G. Frasca-Caccia, F. Iavernaro

TL;DR

The paper develops arbitrarily high-order energy-conserving integrators for constrained Hamiltonian systems with quadratic holonomic constraints by extending the Hamiltonian Boundary Value Method (HBVM) within a line-integral framework. It builds polynomial approximations on each time step using shifted Legendre polynomials, enforces constraint and hidden constraint conservation, and derives a discrete vector form that yields robust computation of Lagrange multipliers. Theoretical results establish symmetry, convergence orders, and multiplier accuracy, while extensive numerical experiments on pendulum-like and tethered-satellite problems validate energy conservation and practicality, including non-quadratic potentials where higher k controls energy accuracy. Overall, the approach offers a scalable, high-precision tool for long-time simulation of constrained Hamiltonian dynamics, with potential extensions to general holonomic constraints.

Abstract

In this paper, we define arbitrarily high-order energy-conserving methods for Hamiltonian systems with quadratic holonomic constraints. The derivation of the methods is made within the so-called line integral framework. Numerical tests to illustrate the theoretical findings are presented.

Arbitrarily high-order energy-conserving methods for Hamiltonian problems with quadratic holonomic constraints

TL;DR

The paper develops arbitrarily high-order energy-conserving integrators for constrained Hamiltonian systems with quadratic holonomic constraints by extending the Hamiltonian Boundary Value Method (HBVM) within a line-integral framework. It builds polynomial approximations on each time step using shifted Legendre polynomials, enforces constraint and hidden constraint conservation, and derives a discrete vector form that yields robust computation of Lagrange multipliers. Theoretical results establish symmetry, convergence orders, and multiplier accuracy, while extensive numerical experiments on pendulum-like and tethered-satellite problems validate energy conservation and practicality, including non-quadratic potentials where higher k controls energy accuracy. Overall, the approach offers a scalable, high-precision tool for long-time simulation of constrained Hamiltonian dynamics, with potential extensions to general holonomic constraints.

Abstract

In this paper, we define arbitrarily high-order energy-conserving methods for Hamiltonian systems with quadratic holonomic constraints. The derivation of the methods is made within the so-called line integral framework. Numerical tests to illustrate the theoretical findings are presented.
Paper Structure (15 sections, 16 theorems, 111 equations, 4 figures, 3 tables)

This paper contains 15 sections, 16 theorems, 111 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Polynomial approximation
  3. Constraints conservation
  4. Energy conservation
  5. Computing the approximate Lagrange multipliers
  6. Accuracy
  7. Hidden constraints and Lagrange multipliers
  8. Discretization
  9. Discrete problem in vector form
  10. Numerical examples
  11. The simple pendulum
  12. A modified pendulum
  13. A conical pendulum
  14. A tethered satellite system
  15. Conclusions

Key Result

Theorem 1

The vector $\lambda$ exists and is uniquely determined, provided that (regular) holds true. In fact, in such a case, from (lambda1), we obtain

Figures (4)

  • Figure 1: phase portrait (left) and Lagrange multiplier (right) for problem (\ref{['pend']})--(\ref{['pend0']}).
  • Figure 2: phase portrait (left) and Lagrange multiplier (right) for the problem defined by (\ref{['pendHmod']}), with $q^*=(2,0)^\top$, using the initial values (\ref{['pend0']}); in the left-plot, the star denotes the second charged point.
  • Figure 3: phase portrait the problem (\ref{['cpend']})--(\ref{['cpend0']}).
  • Figure 4: tethered satellite system described by problem (\ref{['tetH']})--(\ref{['p0']}).

Theorems & Definitions (20)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • Theorem 4
  • Lemma 2
  • Lemma 3
  • Theorem 5
  • ...and 10 more