Does anyone knows why it is impossible to obtain an odd order equation of motion from a Lagrangian? I recently heard that, but I can't find a nice discussion anywhere. It is also interesting to note that this might not even valid for field theory, take as an example the Schrodinger equation which can indeed be retrieved from a Lagrangian and is of odd order (because of the first order time derivative), anyone has an idea of why this happens? I mean why we can't get an odd order equation of motion for discrete system with n degrees of freedom but we can get it for some field (which obviously has infinite degrees of freedom). Thanks in advance.