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.

💡 Research Summary

The paper investigates both continuous and discrete Euler‑Lagrange equations that arise from a quadratic Lagrangian, with a focus on the convergence of the discrete solutions toward the continuous ones as the discretisation step ε tends to zero. The authors begin by introducing a non‑local “scale derivative” ∇ε, defined as a weighted sum of 2N+1 forward and backward shifted values of the trajectory x(t). Replacing the ordinary time derivative ẋ in the Lagrangian L(x, ẋ) by ∇ε x yields a discrete action functional whose critical points satisfy a discrete Euler‑Lagrange (D.E.L.) system. The continuous counterpart (C.E.L.) is the usual second‑order linear differential equation obtained from the classical variational principle.

Both Lagrangians are taken to be quadratic forms \