Discovering Liouville Integrability in Classical Mechanics

[baxter]

Hi

Let a classical particle with unit mass subjected to a radial potential V and moving in a plane.

The Hamiltonian is written using polar coordinates [itex](r,\phi)[/itex]

[itex]H(r,\phi) = \frac{1}{2}(\dot{r}^2+r^2\dot{\phi}^2) - V(r)[/itex]

I consider the angular momentum modulus [itex]C=r^2\dot{\phi}[/itex],
and I want to show that the system is Liouville integrable (the problem is planar so I have to find a first integral (that is ,C) which commute to the Hamiltonian).

My question is : when I want to compute the Poisson bracket [itex]{H,C}[/itex], the only variable is r and [itex]\phi[/itex] ? Because the conjuguate variables [itex]\dot{r}[/itex] and [itex]\dot{\phi}[/itex] appeared in H...
So I should write [itex]H(r,\phi,\dot{r},\dot{\phi})[/itex] and compute the partial derivative with respect to these four variables ?

Thanks

 

[baxter]

somes explanations ?

 

[baxter]

up up up

 

[vanhees71]

In Hamiltonian dynamics you must substitute the generalized velocities with the canonical momenta. Otherwise the formalism doesn't work properly. There's an alternative formalism, where you only perform the Legendre transformation from the Lagrange to the Hamilton formalisms for some generalized coordinates, which goes under the name of Routh, but I've never seen an application of it in practice (if you are interested in that, see A. Sommerfeld, Lectures on Theoretical Physics, Vol. 1).

In your case the Lagrangian should read
[tex]L=\frac{m}{2} (\dot{r}^2+r^2 \dot{\phi}^2) - V(r).[/tex]
The canonically conjugated momenta thus are
[tex]p_r=\frac{\partial L}{\partial \dot{r}}=m \dot{r}, \quad p_{\phi}=\frac{\partial L}{\partial \dot{\phi}}=m r^2 \dot{\phi}.[/tex]
The Hamiltonian now is
[tex]H(q,p)=\dot{q} \cdot p-L=\frac{m}{2} (\dot{r}^2+r^2 \dot{\phi}^2) +V(r) = \frac{1}{2m} \left (p_r^2 +\frac{p_{\phi}^2}{r^2} \right ) + V(r).[/tex]
Now you have two first integrals:

(a) the Hamiltonian doesn't depend explicitly on time. Thus it is conserved:
[tex]H=E=\text{const}.[/tex]
(b) the variable [itex]\phi[/itex] is cyclic, i.e., the Hamiltonian doesn't depend on it. Thus the canonical momentum conjugate to this variable is conserved too:
[tex]p_{\phi}=\text{const}.[/tex]
Thus you have two integral of motion for a system with two degrees of freedom. It is thus integrable.

 

[PJC01]

for your question! It is great to see that you are exploring the concept of Liouville integrability in classical mechanics. To answer your question, when computing the Poisson bracket, you are correct in including all four variables (r, \phi, \dot{r}, \dot{\phi}) in the Hamiltonian. This is because the Poisson bracket is defined as the partial derivative of one variable with respect to another, and in this case, we are interested in finding the Poisson bracket between the Hamiltonian and the angular momentum modulus. Therefore, both the radial and angular variables, as well as their conjugate momenta, should be included in the calculation.

Furthermore, the fact that the Hamiltonian contains the conjugate momenta \dot{r} and \dot{\phi} does not affect the computation of the Poisson bracket. These variables are still considered variables in the Hamiltonian, and their presence does not change the fact that r and \phi are the only variables present in the Poisson bracket calculation.

In summary, when computing the Poisson bracket between the Hamiltonian and the angular momentum modulus, all four variables (r, \phi, \dot{r}, \dot{\phi}) should be included in the Hamiltonian, and the partial derivative should be taken with respect to all four variables. I hope this helps clarify your question and good luck with your exploration of Liouville integrability!