Discrete Calculus of Variations for Quadratic Lagrangians. Convergence Issues

TL;DR

This paper investigates the convergence properties of continuous and discrete Euler-Lagrange equations derived from quadratic Lagrangians, focusing on numerical schemes and the harmonic oscillator case.

Contribution

It introduces a matrix-based framework for solving quadratic Lagrangian equations and analyzes convergence issues, especially in non-resonant and harmonic oscillator scenarios.

Findings

Unconditional convergence does not hold for the harmonic oscillator.

A matrix framework facilitates solving quadratic Lagrangian equations.

Convergence depends on oscillatory and non-resonance conditions.

Abstract

We study in this paper the continuous and discrete Euler-Lagrange equations arising from a quadratic lagrangian. Those equations may be thought as numerical schemes and may be solved through a matrix based framework. When the lagrangian is time-independent, we can solve both continuous and discrete Euler-Lagrange equations under convenient oscillatory and non-resonance properties. The convergence of the solutions is also investigated. In the simplest case of the harmonic oscillator, unconditional convergence does not hold, we give results and experiments in this direction.