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In my book, it says that Hamilton's equations of motion are equations of the first order in the time and that they describe the motion of the system in the $2S$-dimensional phase space. Could someone explain clearly what this means, and what exactly a phase space is?

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If we have a set of $S$ generalized coordinates $q_i$ along with the corresponding conjugate momentum $$p_i=\frac{\partial L}{\partial \dot {q_i}}$$

then we can obtain the Hamiltonian $$H=\sum_ip_i\dot q_i-L$$

And the following equations can be obtained: $$\dot p_i=-\frac{\partial H}{\partial q_i}$$ $$\dot q_i=\frac{\partial H}{\partial p_i}$$

The phase space is just a term used to describe all "coordinates" $\left(q_1,q_2,...,q_S,p_1,p_2,...,p_S\right)$. Therefore, the phase space consists of $2S$ dimensions ($S$ generalized coordinates and $S$ conjugate momenta), and we have $2S$ first order equations in time to describe our trajectories in this phase space.

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