Integrable system

Remarks

Clause 1 with $i = 1$ says that each $F_j$ is a first integral of the system, since $\frac{d}{dt} F_j = \{F_j, H\}$ along solutions. Functions that satisfy clause 1 are said to be in involution.

Example: on $\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, let $H = \frac{1}{2}(p_1^2 + q_1^2) + \frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \frac{1}{2}(p_2^2 + q_2^2)$. Then $\{H, F_2\} = \{\frac{1}{2}(p_2^2 + q_2^2), \frac{1}{2}(p_2^2 + q_2^2)\} = 0$, since the first summand of $H$ does not depend on $q_2, p_2$. The differentials $dH$ and $dF_2$ are independent wherever $(q_1, p_1) \ne (0, 0)$ and $(q_2, p_2) \ne (0, 0)$, an open dense set.

Non-example: the same $H$ with $F_2 = H$ satisfies clause 1 but not clause 2, since $dF_1 = dF_2$ everywhere.

Every Hamiltonian system with one degree of freedom ($n = 1$) is integrable as soon as $dH \ne 0$ on an open dense set. The Liouville–Arnold theorem describes the motion of an integrable system near a compact regular level set.