Exercise 1.4: Lagrangian actions
This exercise has us calculating the actual value of the action along some realizable path taken by a free particle.
formula not implementedFor a free particle an appropriate Lagrangian is
formula not implementedSuppose that x is the constant-velocity straight-line path of a free particle, such that inline_formula not implemented and inline_formula not implemented. Show that the action on the solution path is
I'm not sure I see the point of this exercise, for developing intuition about Lagrangian mechanics. I think it may be here to make sure we understand that we're not minimizing the function inline_formula not implemented. We're minimizing (finding the stationary point of) the integral of inline_formula not implemented between inline_formula not implemented and inline_formula not implemented.
The velocity is constant between the two points, so it must be equal to the difference in position over the difference in time:
formula not implementedThe action is equal to the integral of inline_formula not implemented evaluated between the two time points:
formula not implementedSubstitute in and simplify:
formula not implementedBoom, solution achieved.