Numerical Calculation of the Orbital Invariant of Goryachev—Chaplygin and Lagrange Systems
This chapter presents numerical calculations of the orbital invariant that classifies integrable Hamiltonian systems on constant-energy surfaces. The Bolsinov-Fomenko theory of orbital classification of integrable Hamiltonian systems is finer than that of fiber-classification, but it is difficult to obtain such an invariant without computers. The orbital invariant W* t consists of some components that can be found and investigated analytically, except for the rotation vector. The rotation vector is an orbital invariant based on the rotation function, which is defined on a family of Liouville tori (the exact definition is given below). Here we consider integrable Hamiltonian systems whose rotation functions have an explicit analytic form: the Goryachev-Chaplygin and Lagrange systems defined in the preceding chapter.
KeywordsHamiltonian System Bifurcation Diagram Action Variable Rotation Vector Critical Curve
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