How cyclic coordinates affect the dimension of the cotangent manifold

[mjordan2nd]

Our professor's notes say that "In general, in Hamiltonian dynamics a constant of motion will reduce the dimension of the phase space by two dimensions, not just one as it does in Lagrangian dynamics." To demonstrate this, he uses the central force Hamiltonian,

$$H=\frac{P_r^2}{2m}+\frac{p_{\theta}^2}{2mr^2}+ \frac{p_{\phi}}{2mr^2 sin^2 \theta} + V(r).$$

Since by Hamilton's equation $\dot{p_{\phi}}=0$ this is a constant of the motion. So specifying $p_{\phi}=\mu$ gives us a 5 dimensional manifold. The notes go on to state that, "Furthermore, on each invariant submanifold the Hamiltonian can be written

$$H=\frac{P_r^2}{2m}+\frac{p_{\theta}^2}{2mr^2}+ \frac{\mu}{2mr^2 sin^2 \theta} + V(r),$$

which is a Hamiltonian involving only two freedoms $r$ and $\theta$. Therefore the motion actually occurs on a 4-dimensional submanifold of the 5-dimensional submanifold of $T^*Q$ . . ." However, to me it looks like we still have five degrees of freedom: $p_{\theta}, p_r, r, \theta,$ and $\phi$. So I'm not sure what he means when he says that the presence of a constant of motion reduces the dimension of the cotangent manifold by 2. Is he saying that if w specify a numerical value for H then the dimension is reduced from 5 to 4, or does just the presence of a cyclic coordinate reduce the dimension from 6 to 4?

 

[ChrisVer]

but isn't you Φ fixed?

 

[dextercioby]

Check out corollary 2, page 67 from V.I. Arnold's <Mathematical Methods of Classical Mechanics>. The phase space dimension is indeed reduced by 2xn, n=nr. of cyclic coordinates for the Hamiltonian.

 

[mjordan2nd]

Thank you. I think I'm just confusing terminology.

 

[PJC01]

I would explain that the dimension of the cotangent manifold is directly related to the number of degrees of freedom in a system. In Hamiltonian dynamics, a constant of motion is a quantity that remains constant throughout the motion of the system. This means that it is independent of time and does not change as the system evolves.

In the example given, the constant of motion is p_{\phi}=\mu, which is the momentum in the azimuthal direction. This means that for any given value of p_{\phi}, the other momenta (p_{\theta} and p_r) and the position coordinates (r and \theta) can vary independently. This leads to a 5-dimensional manifold, as there are five independent variables that can vary.

However, on each invariant submanifold where p_{\phi} is fixed, the Hamiltonian can be rewritten as a function of only two variables, r and \theta. This means that the motion of the system actually occurs on a 4-dimensional submanifold of the 5-dimensional cotangent manifold. This is because one degree of freedom (p_{\phi}) is eliminated due to the presence of the constant of motion.

Therefore, the presence of a constant of motion reduces the dimension of the cotangent manifold by 2, from 5 to 3. This means that there are only four independent variables that can vary, instead of the original six (p_{\theta}, p_r, r, \theta, \phi, and t).

In summary, the reduction in dimension occurs because the constant of motion eliminates one degree of freedom, and on each invariant submanifold, the Hamiltonian can be rewritten as a function of only two variables, further reducing the dimension by one. This is a fundamental concept in Hamiltonian dynamics and is important in understanding the behavior of systems with cyclic coordinates.