Discrete Calculus of Variations for Quadratic Lagrangians. Convergence Issues

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.