Consider the following Lagrangian in Cartesian coordinates:
L(x, y, x', y') = 12 (x^ 2 + y^2) -sqrt(x^2 + y^2)
(a) Write the Lagrange equations of motion, and show that x = cos(t);
y =sin(t) is a solution.
(b) Changing from Cartesian to polar coordinates, x = r cos ; y = r sin ,
show that the Lagrangian becomes
L(r;θ ; r';') = 1/2 (r'^2 + r^2θ^2)- r;
and hence find the Lagrange equations in polar coordinates.
(c) Show that the solution given in Cartesian coordinates in (a) is still a
solution when expressed in polar coordinates.
The Attempt at a Solution
got both as i think there is 2 degrees of freedom
put in x=cost, y=sint
put these it above i get
for a this is where i think i get stuck.
and thats where i get stuck. This is a practise exam question, and i want to make sure im going the right way about it. Any help would be much appropriated