Adelphic potentials

Details Adelphic potentials Adelphic potentials

May 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986a%26a...160..107b&link_type=abstract

Astronomy and Astrophysics (ISSN 0004-6361), vol. 160, no. 1, May 1986, p. 107-110.

Statistics

Computation

3

Orbits, Potential Theory, Computational Astrophysics, Partial Differential Equations

Scientific paper

All orbits which a material point can trace, under all possible initial conditions, in an autonomous conservative field U = A(x,y), constitute a three-parametric family of planar curves. In general, two arbitrary potentials U = A(x,y) and U = B(x,y) are not, of course, expected to have any orbits in common. The present paper examines the conditions under which two given potentials U = A(x,y) and U = B(x,y) have in common a monoparametric family of orbits f(x,y)= c, not given in advance. Such potentials are defined as adelphic. It is shown that if the given functions A and B are functionally independent and if they satisfy a certain condition, then they are adelphic and the corresponding family f(x,y) = c is, in general, determined uniquely. If A and B are functionally dependent and if A satisfies a certain condition, then so does B and the potentials are adelphic, having in common a family f(x,y) = c which again is determined uniquely. In this latter case, f(x,y) = c is a family of isotach orbits. A partial differential equation giving all possible planar isotach orbits is also derived.

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