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Various Aspects of Integrable Hamiltonian Systems

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Book cover Dynamical Systems

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 78))

Abstract

(a) In these informal lecture notes we discuss a number of integrable Hamiltonian systems which have surfaced recently in very different connections. It is our goal to discuss various aspects underlying the integrability of a system like that of group representation, isospectral deformation and geometrical considerations. Since this subject is still far from being understood or being systematic we discuss a number of examples which are seemingly disconnected. In fact, there are some rather unexpected connections like between the inverse square potential of Calogero (Section 4) and the Korteweg de Vries equation. Here we show a surprising new connection between the geodesics on an ellipsoid and Hill's equation with finite gap potential.

(b) The differential equations of mechanics can be written in Hamiltonian form

$${{\dot x}}_{\text{k}} = \frac{{\partial {\text{H}}}}{{\partial {\text{y}}_{\text{k}} }},\,\,\,\,\,\,\,{{\dot y}}_{\text{k}} = - \frac{{\partial {\text{H}}}}{{\partial {\text{x}}_{\text{k}} }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{k = 1,2,}}...{\text{n}}} \right)$$
(1)

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  1. Courant Institute of Mathematical Sciences, New York University, New York, USA

    J. Moser

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  1. J. Moser
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C. Marchioro

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Moser, J. (2010). Various Aspects of Integrable Hamiltonian Systems. In: Marchioro, C. (eds) Dynamical Systems. C.I.M.E. Summer Schools, vol 78. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13929-1_3

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