Cyclic integral? Can anyone explain me what is cyclic integral and give me some instructions how to show that there exist cyclic integral for Lagrangian
$$L~=~\frac{1}{2}ma^2{\dot {\theta}}^2+\frac{1}{2}ma^2{\dot {\phi}}^2\sin^2\theta+mga\cos\theta?$$  
 A: Given a Lagrangian $L(q^1, \dots, q^N, \dot q^1, \dots, \dot q^N)$ defined on coordinates $q_1, \dots q_N$ and their corresponding velocities $\dot q_1, \dots, \dot q_N$, a cyclic coordinate $q^{i}$ is one on which the Lagrangian doesn't explicitly depend;
$$
  \frac{\partial L}{\partial q^{i}} = 0
$$
Since the Euler Lagrange equations for each coordinate $q^k$ is
$$
  \frac{d}{dt}\frac{\partial L}{\partial \dot q^k} = \frac{\partial L}{\partial q^k}
$$
The Euler Lagrange equation for the cyclic coordinate $q^i$ has vanishing right hand side and becomes
$$
  \frac{d}{dt}\frac{\partial L}{\partial \dot q^i} = 0
$$
so that the quantity (incidentally called the canonical momentum corresponding to the coordinate $q^i$)
$$
  p_i=\frac{\partial L}{\partial \dot q^i}
$$
is conserved.  Such a conserved quantity is often referred to as a first integral, and although I've never personally seen the term "cyclic integral," I'm guessing that it is simply the conserved quantity generated from the existence of a cyclic coordinate as described above.
In the Lagrangian you have written down, the coordinate $\phi$ is cyclic since it doesn't appear explicitly in the Lagrangian.  As a result the quantity $\partial L/\partial \dot\phi$ is conserved, and this is probably the "cyclic integral" you are looking for.
