A Nonlinear PDE for Families of Orbits on a Given Surface

Abstract

AbstractWe study a nonlinear PDE which descibes monoparametric families of orbits on a certain surface produced by two-dimensional potentials. We face the following version of the direct problem of Newtonian Dynamics: Given a surface Sand a two-dimensional potential$$V = V(u,v)$$ V = V ( u , v ) , determine all the isoenergetic families of orbits$$f(u,v) = $$ f ( u , v ) = c ($$c = const.$$ c = c o n s t . ), that is, families of orbits which are traced by a test particle with the same preassigned value of the total energy $${\mathcal {E}} = {\mathcal {E}}_{0}$$ E = E 0 . We are interested especially in those orbits which are described by energy $${\mathcal {E}}_{0}$$ E 0 = 0. Thus, using Merten’s equation (ZAMM 61:252–253, 1981), we establish a new, nonlinear PDE for the “slope function” $$\gamma $$ γ = $${\frac{{f_{v}}}{{f_{u}}}}$$ f v f u which represents well the corresponding family of orbits $$f(u,v) = c$$ f ( u , v ) = c on the given surface S. We find two necessary and sufficient differential conditions, one for the potential V = V (u, v) and another one for the slope function $$\gamma $$ γ , so that the above PDE has solution. Furthermore, we determine the general solution of the above PDE. Not only real but also complex potentials can produce these families of orbits on the given surface S. Several examples are offered.