  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.
