Liouville-Arnold theorem

proper noun

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

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Meanings

proper nounthe Liouville-Arnold theorem
  1. In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

Origin

Named after Joseph Liouville and Vladimir Arnold.

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Cite this entry
APA

Liouville-Arnold theorem. (n.d.). In WatkinsX Dictionary. Retrieved November 5, 2026, from https://dictionary.watkinsx.com/word/Liouville-Arnold%20theorem

MLA

“Liouville-Arnold theorem.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/Liouville-Arnold%20theorem. Accessed November 5, 2026.

Chicago

WatkinsX Dictionary, s.v. “Liouville-Arnold theorem,” accessed November 5, 2026, https://dictionary.watkinsx.com/word/Liouville-Arnold%20theorem.

Source: Liouville-Arnold theorem on Wiktionary, CC BY-SA 4.0.