Consider the following discussion
I am struggling to understand what is going on. For example, I do not understand why $$ \frac{d H}{d t}=g_{2} \dot{x}-g_{1} \dot{y}. $$ I thought it could be via the chain ruleagain, but $\partial H / \partial x \neq g_2$.
Also, I don't understand the conclusion. In particular, if an orbit is periodic and closed, why is the given integral $0$?
I would appreciate any help with this.