Isovortical Orbits in Uniformly Rotating Coordinates
Abstract
For the conservative, two degreeoffreedom system with autonomous potential functionV(x,y) in rotating coordinates; <MediaObject> <ImageObject FileRef="10569_2005_Article_BF01234309_TeX2GIFE1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"/> </MediaObject> dot u  2n\upsilon = V_x , dot \upsilon + 2nu = V_y , vorticity (v _{ x }u _{ y }) is constant along the orbit when the relative velocity field is divergencefree such that: <MediaObject> <ImageObject FileRef="10569_2005_Article_BF01234309_TeX2GIFE2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"/> </MediaObject> u(x,y,t) = ψ _y , \upsilon (x,y,t) =  ψ _x . Unlike isoenergetic reduction using the Jacobi, integral and eliminating the time,nonsingular reduction from fourth to secondorder occurs when (u,v) are determined explicitly as functions of their arguments by solving for ψ (x, y, t). The orbit function ψ satisfies a secondorder, nonlinear partial differential equation of the Monge Ampere type: <MediaObject> <ImageObject FileRef="10569_2005_Article_BF01234309_TeX2GIFE3.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"/> </MediaObject> 2(ψ _{xx} ψ _{yy}  ψ _{xy}^2 )  2(ψ _{xx} + ψ _{yy} ) + V_{xx} + V_{yy} = 0 . Isovortical orbits in the rotating frame arenot level curves of ψ because it contains time explicitly due to coriolis effects. Rather, (x, y) coordinates along the orbit are obtained, from (u, v) either by numerical integration of the kinematic equations, or by partial differentiation of the Legendre transform ⋏ of ψ. In the latter case, ⋏ is shown to satisfy a nonlinear, secondorder partial differential equation in three independent variables, derived from the MongeAmpere Equation. Complete reduction to quadrature is possible when spacetime symmetries exist, as in the case of central force motion.
 Publication:

Celestial Mechanics
 Pub Date:
 September 1986
 DOI:
 10.1007/BF01234309
 Bibcode:
 1986CeMec..39..249H