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My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ given as integrals, for given initial conditions $(q(0),p(0))$, of algebraic functions in the hamiltonian $H(p,q)$. (In a broad sense that is, for instance derivatives of polynomials should be considered algebraic operators.) In particular computing approximate solutions of completely integrable hamiltonian systems can be carried out in polynomial time in the time variable, with an oracle for the computation of the hamiltonian to any given finite accuracy (maybe some more conditions on the hamiltonian should be imposed).

More generally the notion of closed form solution should correspond to "relatively easily computable". Or at least computable using well-known functions (e.g. elementary, elliptic, etc.).

Are there articles giving and studying such general definitions of closed forms? Are there articles studying the link between complexity, closed forms, and "chaos"?

Thank you.

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