Translational symmetries of quadratic Lagrangians

In this paper we show that a quadratic lagrangian, with no constraints, containing ordinary time derivatives up to the order $m$ of $N$ dynamical variables, has $2mN$ symmetries consisting in translation variables solutions equations motion. We construct explicitly generators these transformations and prove they satisfy Heisenberg algebra. also analyse other specific cases which are not included our previous statement: Klein-Gordon Fermi oscillators Dirac lagrangian. first case, system is described by an equation involving partial derivatives, second case Grassmann third shows both features. Furthermore, oscillator field lagrangians lead class constraints. last two there translational algebra generators. For find continuum version algebra, whereas cases, satisfy, after quantization, creation annihilation operators.