Some new discretizations of the Euler–Lagrange equation

•In this work, we present some discrete approximations of the continuous Euler–Lagrange equation.•The new approximations are obtained without using variational techniques.•As far as we know, the proposed approach is the first attempt in this direction.•Comparison with other existing approaches are provided, showing a good performance. The Veselov approach provides a discrete formulation of the Euler–Lagrange equation. To get this, a discrete Lagrangian version of a continuous one is considered and then a variational process is used. This problem has been studied in many papers by different authors, according to references and therein citations. This type of discretization can be useful in the case when the continuous Euler–Lagrange equation is given in a semispray form, which is difficult to solve effectively (as for example in the many-body problem). Our aim is to consider a given continuous Lagrangian and to construct directly discrete approximations of the corresponding Euler–Lagrange equation. This is done without considering a discrete Lagrangian and a variational process, nor by using a difference equation of geodesics. Some numerical examples are included in order to compare the performance of the proposed approximations versus the classical Veselov approach.