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We say that two dynamical variables $f$ and $g$ are in involution if their Poisson bracket vanishes, i.e., $\{f,g\}=0$. Why is it called involution?

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An involution is, according to the OED, "A function or transformation that is equal to its inverse."

Whittaker discusses involution systems in his A Treatise on the Analytic Dynamics of Particles and Rigid Bodies p. 322:

147. Invoultion systems.

Let $(u_1,, u_2 , …, u_r)$ denote $r$ functions of $n$ independent variables $$(q_1, q_2 , …, q_n , p_1, …, p_n);$$if it is possible to express all the Poisson-brackets $(u_i,u_k)$ as functions of $(u_1,u_2,…,u_r)$, the functions $(u_1,u_2,…,u_r)$ are said to form a function-group*. Any function of $(u_1,u_2,…,u_r)$ belongs to this group.

If the quantities $(u_i,u_k)$ are all zero, the functions $(u_1,u_2,…,u_r)$ are said to be in involution, or to form an involution-system.


*Lie, Math. Ann. VIII. (1875). p. 215.

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