Consider a particle of mass m that is constrained to move on a cone defined by z? = a2(x2 + y2) (a is a constant) with no external force other than the constraint on the particle. (take z> 0)
a) write down the lagrangian in cartesian coordinates with a lagrange multiplier that imposes the constraints.
b) find the euler-lagrange equations in cartesian coordinates.
c) rewrite the lagrangian cylindrical polar coordinates. solve the constraint for z and substitute the solution into the lagrangian to get a lagrangian l(r, r, φ).
d) find the euler-lagrange equations that follow from l(r, r, 0) that you found. z-component of the angular momentum