We take a general Hamiltonian system $$ H(\boldsymbol x_1,\cdots,\boldsymbol x_n,\boldsymbol v_1,\cdots,\boldsymbol v_n) = \frac{1}{2}\sum_{i=1}^nm|\boldsymbol v_i|^2+\sum_{1\leq i < j \leq n}U(|\boldsymbol x_i-\boldsymbol x_j|)+V(\boldsymbol x_i) $$ where $\boldsymbol x_i, \boldsymbol v_i\in \mathbb{R}^3$, $U:\mathbb{R^+}\to\mathbb{R}$ is the interacting potential (like Coulomb potential or Lennard-Jones potential), and $V:\mathbb{R}^3\to\mathbb{R}$ is a potential trapping the particles into a finite region(like a harmonic oscillator potential, or square-well potential).
The question is: is such a system ergodic? Or in other words, is the trajectory of this system dense in the level set of the Hamiltonian function? Or at least, can we prove that there is no other conserved quantity independent of Hamiltonian?
Sure this question is very general, but I just want to know how much people have already known about this.