Mathematics – Dynamical Systems
Scientific paper
May 1985
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1985cemec..36....1h&link_type=abstract
Celestial Mechanics (ISSN 0008-8714), vol. 36, May 1985, p. 1-18.
Mathematics
Dynamical Systems
3
Dynamical Systems, Equations Of Motion, Monge-Ampere Equation, Vorticity, Degrees Of Freedom, Kepler Laws
Scientific paper
For an autonomous, conservative, two degree-of-freedom dynamical system, vorticity is constant along the orbit if the velocity field is divergence-free such that: u(x, v) = ψy, v(x, y) = -ψx. Isovortical orbits in configuration space are level curves of a scalar autonomous function ψ(x, v) satisfying a differential equation of the Monge-Ampère type: 2(ψxxψyy-ψxy2)+Uxx+Uyy = 0, where U(x, y) is the autonomous potential function. The solution for the time variable is reduced to a quadrature following determination of ψ. Self-similar solutions of the Monge-Ampère equation under Birkhoff's one-parameter transformation group are derived for homogeneous (power-law) potential functions. It is shown that Keplerian orbits belong to the class of planar isovortical flows.
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