I was introduced the generator of Galilean boost $K=mx-pt$.

I was given an Hamiltonian with several particles: $H=\sum_i \frac{p_i^2}{2m_i}+V(|x_i-x_j|)$ where the potential only depends on the relative position between the particles.

I was asked to compute the poisson brackets $\{K,H\}$ and show that the general time evolution formula is given by : $$\frac{d K}{d t}=\frac{\partial K}{\partial t}+\left\{K, H\right\}.$$

I managed to derive the time evolution formula but I feel I am doing something wrong. First the Poisson brackets: \begin{align} \{K,H\}&= \frac{\partial K}{\partial x}\frac{\partial H}{\partial p}-\frac{\partial K}{\partial p}\frac{\partial H}{\partial x}\\ &=p+t\frac{ \partial V}{ \partial x}\\ &=p-t\frac{d p}{d t}.\\ \end{align}

I was told that $\frac{d K}{d t}=0$, which I do not understand why. I found: $$\frac{d K}{d t}=-t\frac{d p}{d t}\;(\,\neq 0\;?)$$

Finally, I read that the Poisson bracket $\{K,H\}$ describes the effect of an infinitesimal boost on the system’s energy. This eludes my understanding. I could show that the Hamiltonian changes when I perform a change of coordinates $x'=x-v_0t$, but I do not know how to link it with the Poisson brackets.

I find this an interesting topic, and if someone could help clarify my understanding with some explanations I would be very grateful. Any remark or link to references is always appreciated.


I'm not sure WP would help you.

Your comment clarification is crucial. It indicates your potential is invariant under a translation, whether that translation is proportional to the time, or not. Without loss of generality, take two particles, at positions x and y. You then have $$ (\partial_x + \partial_y) V(|x-y|)= ((\partial_x + \partial_y)|x-y|) ~~V'(|x-y|) =0, $$ since the left parenthesis vanishes, $$ (\partial_x + \partial_y)\sqrt {(x-y)^2}=0. $$ That is, for your expression K really indicating $\sum_i K_i$ as well as the PBs, you get $$ \partial_t K + t \{ K,H\}= t(\partial_x + \partial_y) V(|x-y|)=0. $$

An infinitesimal transformation generated by K is $$ \delta x_i = v_0 \{ x_i , K \}= -v_0 t, \qquad \delta p_i = v_0 \{ p_i , K \}= -mv_0 \\ \leadsto ~~~ \delta H = -v_0 \sum_i (p_i+ t\partial_i V) = -v_0 \sum_i p_i . $$ This reflects the corresponding term in the WP Lie algebra, involving the Hamiltonian and the common Galilean boost.

Note the invariant $mH-P^2/2$, so that $$\delta (mH-P^2/2) =0.$$

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