Different Lagrangians can lead to the same equations of motion. In particular, w

Question

Different Lagrangians can lead to the same equations of motion. In particular, when we add to the Lagrangian the total time derivative of a function of the coordinates, the action changes only by boundary terms. Since the equations of motion correspond to variations of the action with fixed boundary conditions, they should be unaffected by such total derivative terms. This problem makes this explicit. Consider two Lagrangians: L and L' =L+ df/dt where f({q i},t) is an arbitrary function of the coordinates and time. Express df/dt in terms of functions of qi, q i and t (that is, do the derivative !). Find the Lagrange equation for L' and show that it is the same as that of L.

Explanation / Answer

a) so df/dt full derivative = in terms of partials = df/dqi dqi/t + df/dt (all these are partials) = df/dqi qi' + df/dt b) dL'/dqi' = dL/dqi' + df/dqi d/dt(dL'/dqi') = d/dt dL/dqi + d/dt(full deriv) df/dqi =d/dt dL/dqi + d/dt df/dqi +d^2f/dqi^2 qi' dL'/dqi = dL/dqi + d^2f/dqi^2 qi' + d/dqi df/dt so Lagrange equation dL'/dqi = d/dt dL/dqi' dL/dqi + d^2f/dqi^2 qi' + d/dqi df/dt=d/dt dL/dqi' + d/dt df/dqi +d^2f/dqi^2 qi' terms cancel and we are left with dL/dqi = d/dt dL/dqi'

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