Hi all, I've recently been asked for an explanation as to why the Lagrangian is a function of the positions and velocities of the particles constituting a physical system. What follows is my attempt to answer this question. I would be grateful if you could offer your thoughts on whether this is a correct description, and if it is not, please could you enlighten me. Thanks for your time. The Lagrangian characterizes the state of a system at each point along an arbitrary configuration path between two fixed configurations (arbitrary other than the fact that it must satisfy the boundary conditions). At each point along the configuration path we can give a local description of the path by specifying the coordinate values at that point as well as their first-order (and, in general, second-order and higher-order) derivatives at that point (i.e. a Taylor expansion about that point). The state of the system at a given point along the configuration path should depend on no more than the local description of the path around that point. This implies that the Lagrangian should be a function of the coordinate values, and their derivatives (along with higher order derivatives) at that point. The action for the given path is then constructed by integrating the value of the Lagrangian at each point along the configuration path within the specified time interval. The classical equations of motion are second-order and therefore, as the principle of stationary action asserts that the actual physical configuration path taken by the system between the two fixed configurations (and hence the one in which the state of the system characterized by the Lagrangian at each point along the path has any physical meaning) is the one which gives a stationary value to the action, this implies that the Lagrangian should be, at most, first-order and hence a function of the positions and velocities of the particles within the system (and hence, its variation will be second order).