Does the Hamiltonian Symmetry Extend to All N Dimensions?

[Bamsen]

Homework Statement

I shall consider the Hamilton

H(q,p)=1/2\sum{p_l^2}+1/2\sum{q_l^2}+\gamma(\sum{q_l^2})^2

all the sums if from l=1 to l=N. (gamma is bigger or equal to zero) I shall discuss the symmetry/asymmetry plus determine that the form of the qeneral solution is qualitative similar to the form of the general N=2 solution.

Homework Equations

The Attempt at a Solution

I am a bit helpless so any advice would be helpfull. I tried to look at the equations of motions ...but i all the time get problems with the last coupled term. SO please help...

Homework Statement

 

[BigJDubs]

I shall consider the HamiltonH(q,p)=1/2\sum{p_l^2}+1/2\sum{q_l^2}+\gamma(\sum{q_l^2})^2all the sums if from l=1 to l=N. (gamma is bigger or equal to zero) I shall discuss the symmetry/asymmetry plus determine that the form of the qeneral solution is qualitative similar to the form of the general N=2 solution.Homework Equations The Attempt at a SolutionThe Hamiltonian is symmetric under permutation of any two coordinates, since it is a sum of squares of coordinates and momenta. The equations of motion are given by: \dot{q_l} = \frac{\partial H}{\partial p_l} = p_l\dot{p_l} = -\frac{\partial H}{\partial q_l} = -q_l - 4\gamma \left(\sum_{m=1}^N q_m^2\right) q_lThis is a system of coupled differential equations with no analytical solution in terms of simple functions. However, for N=2, the general solution can be written as follows: q_1(t) = A_1 cos(\omega t + \phi_1) + B_1 sin(\omega t + \phi_2)q_2(t) = A_2 cos(\omega t + \phi_3) + B_2 sin(\omega t + \phi_4)where A_1, A_2, B_1, B_2, $\phi_1$, $\phi_2$, $\phi_3$ and $\phi_4$ are constants of integration. For general N, the general solution will have a similar form, with N sets of parameters instead of just 4.