From ( l = m r^2 \sin^2\alpha \dot{\phi} ), we get ( \dot{\phi} = \frac{l}{m r^2 \sin^2\alpha} ). Substitute: [ \ddot{r} = r \sin^2\alpha \left( \frac{l}{m r^2 \sin^2\alpha} \right)^2 - g\cos\alpha = \frac{l^2}{m^2 r^3 \sin^2\alpha} - g\cos\alpha ] This is the radial equation of motion. The effective potential is ( V_{\text{eff}}(r) = \frac{l^2}{2m r^2 \sin^2\alpha} + mgr\cos\alpha ).
Solving the complex orbits of planets and the dynamics of two-body systems. Introduction To Classical Mechanics Atam P Arya Solutions
This is where many students seek the most help, as it introduces the "Principle of Least Action" and generalized coordinates. From ( l = m r^2 \sin^2\alpha \dot{\phi}
To illustrate the value of proper solutions, let us examine three notoriously difficult sections from Arya where students desperately seek help. Solving the complex orbits of planets and the
( L = T - V = \frac{1}{2}m(\dot{r}^2 + r^2\sin^2\alpha \dot{\phi}^2) - mgr\cos\alpha ).
Make sure your coordinate transformations are solid before starting the physics. Lagrangian and Hamiltonian Dynamics:
Detailed breakdowns of simple, damped, and forced harmonic motion.