Isovortical orbits of autonomous, conservative, two degree-of-freedom dynamical systems

Abstract

For an autonomous, conservative, two degree-of-freedom dynamical system, vorticity (the curl of velocity) is constant along the orbit if the velocity field is divergence-free such that:

$$u\left( {x, v} \right) - \psi _y , v\left( {x, y} \right) = - \psi _x .$$

Isovortical orbits in configuration space are level curves of a scalar autonomous function Ψ (x, v) satisfying a second-order, non-linear partial differential equation of the Monge-Ampere type:

$$2\left( {\psi _{xx} \psi _{yy} - \psi _{xy}^2 } \right) + U_{xx} + U_{yy} = 0,$$

where U(x. y) is the autonomous potential function. The solution Soc the time variable is reduced to a quadrature following determinatio of Ψ. Self-similar solutions of the Monge-Ampere 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.