Hamilton equation

Also, consider when the phase space is $T^{\ast }M$. The physical motion can be described by $\mathrm{\Gamma }:\mathbb{R}\to T^{\ast }Q$ that makes the graph commutative: $\dot{\mathrm{\Gamma }}(t)=\mathrm{\#}dH(\mathrm{\Gamma }(t))$, where on the right is the Hamilton vector field.
ref: the Architecture of Mechanics by Yujin Tong, the following graph is also from him.