Consider the Lagrangian \(L^{\prime}=m \dot{x} \dot{y}-k x y\) for a particle free to move in two dimensions, where \(x\) and \(y\) are Cartesian coordinates, and \(m\) and \(k\) are constants.

(a) Show that his Lagrangian gives the equations of motion appropriate for a two-dimensional simple harmonic oscillator. Therefore as far as the motion of the particle is concerned, \(L^{\prime}\) is equivalent to \(L=(1 / 2) m\left(\dot{x}^{2}+\dot{x}^{2}ight)-(1 / 2) k\left(x^{2}+y^{2}ight)\).

(b) Show that \(L^{\prime}\) and \(L\) do not differ by the total time derivative of any function \(f(x, y)\). Therefore \(L^{\prime}\) is not a member of the class of Lagrangians mentioned in the preceding problems, so there are even more Lagrangians describing a particle than suggested before.