Finding Conserved Quantities for Analytical Mechanics Exam Questions

[JorisL]

Hi all,

I am preparing for my "second chance exam" in analytical mechanics.
It is a graduate course i.e. based on geometry. (Our course notes are roughly based on Arnold's book).

I was able to find some old exam questions and one of those has me stumped, completely.
The question gives 3 general Hamiltonians, no details whatsoever and asks to find as many conserved quantities as possible.
*Warning* The second one really hurts my brain.

$\mathcal{H}_1 = \mathcal{H}_1\left( f_1(q^1,p_1), \ldots, f_N(q^N,p_N)\right)$

(1)​

$\mathcal{H}_2 = g_N( g_{N-1}(\ldots g_2(g_1(q^1,p_1),q^2,p_2)\ldots ,q^{N-1},p_{N-1}),q^N,p_N)$

(2)​

$\mathcal{H}_3 = \sum_{i=1}^N \left(\dot{q}^i(t)\right)^2+V\left( \sum_{i=1}^N \left(q^i(t)\right)^2\right)$

(3)​

Hamiltonian (1) isn't terribly complicated at first sight, however without further information about the functions $f_i$ I'm not sure what I can deduce. As far as I can tell, I cannot claim anything to be conserved except the Hamiltonian itself. (Which is easy to see because there is no explicit time-dependence so $dH_1/dt = \{H,H\} = 0$)

The second Hamiltonian, I don't even understand why the instructor would do such a thing to us. It is horrible and unless someone knows a neat trick or has some cool information about this, I suggest we all pretend that doesn't exist. My guess is that only the conservation of the Hamiltonian follows from this, once again.

Definition (3) seems like a nice expression. But there is a strange thing going on here, we have $\dot{q}^i$ in there but no generalized moments.
I could state that the moments are cyclic but that would be a trap I think.
This Hamiltonian hasn't been "fully transformed" from the Lagrangian.
I believe the "real" Hamiltonian might be invariant under rotations in the coordinates $\vec{q}$.
My (very short and vague) motivation is that the positions appear squared only, no mixing either.

So does anybody have some general remarks/ideas/resources to help me with this kind of stuff?

-Joris

 

[JorisL]

So I made some 'progress', the first 2 are definitely based on the use of (time independent) Hamilton-Jacobi.
Even though this is not much to go by, I suppose it might help some people with the same problems.

I might come back to this later and go into some detail if I can find the time.

 

[samalkhaiat]

Hi all,

$\mathcal{H}_1 = \mathcal{H}_1\left( f_1(q^1,p_1), \ldots, f_N(q^N,p_N)\right)$

(1)​

Calculate
$$\dot{ f }_{ i } = \{ H , f_{ i } \} = \sum_{ n = 1 }^{ N } \left( \frac{ \partial H }{ \partial p^{ n } } \frac{ \partial f_{ i } }{ \partial q_{ n } } - \frac{ \partial H }{ \partial q_{ n } } \frac{ \partial f_{ i } }{ \partial p^{ n } } \right) .$$
Then, use
$$\frac{ \partial f_{ i } }{ \partial q^{ n } } = \frac{ \partial f_{ i } }{ \partial p^{ n } } = 0 , \ \ \mbox{ for all } \ n \neq i .$$

$\mathcal{H}_2 = g_N( g_{N-1}(\ldots g_2(g_1(q^1,p_1),q^2,p_2)\ldots ,q^{N-1},p_{N-1}),q^N,p_N)$

(2)​

Try to calculate
$$\dot{ g }_{ 1 } = \{ H , g_{ 1 } \} .$$

$\mathcal{H}_3 = \sum_{i=1}^N \left(\dot{q}^i(t)\right)^2+V\left( \sum_{i=1}^N \left(q^i(t)\right)^2\right)$

(3)​

Nothing prevent you from setting $\dot{ q }^{ i } = p^{ i }$. Then, you can show that the orbital angular momentum is conserved
$$\frac{ d L_{ i } }{ d t } = \{ H , \epsilon_{ i j k } q^{ j } p^{ k } \} = 0 .$$