- #1
JorisL
- 492
- 189
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)##
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
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