Liouville–Arnold theorem

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In dynamical systems theory, the Liouville–Arnold theorem states 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. The theorem is named after Joseph Liouville and Vladimir Arnold.[1][2][3][4][5](pp270–272)


  1. ^ J. Liouville, « Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853 », JMPA, 1855, p. 137-138, pdf
  2. ^ Fabio Benatti (2009). Dynamics, Information and Complexity in Quantum Systems. Springer Science & Business Media. p. 16. ISBN 978-1-4020-9306-7.
  3. ^ P. Tempesta, P. Winternitz, J. Harnad, W. Miller, Jr., G. Pogosyan, and M. Rodriguez, eds. (2004). Superintegrability in Classical and Quantum Systems. American Mathematical Society. p. 48. ISBN 978-0-8218-7032-7.CS1 maint: uses editors parameter (link)
  4. ^ Christopher K. R. T. Jones; Alexander I. Khibnik, eds. (2012). Multiple-Time-Scale Dynamical Systems. Springer Science & Business Media. p. 1. ISBN 978-1-4613-0117-2.
  5. ^ Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer. ISBN 9780387968902.