Discrete Calculus of Variations for Quadratic Lagrangians. Convergence Issues
For researchers in numerical methods and calculus of variations, this work provides insight into convergence issues of discrete variational schemes, though it is incremental.
The paper studies continuous and discrete Euler-Lagrange equations for quadratic Lagrangians, solving them under oscillatory and non-resonance conditions, but finds that unconditional convergence fails for the harmonic oscillator, supported by theoretical results and experiments.
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.