Abstract
The particular version of the inverse problem of dynamics considered here is: given the ‘slope function’ \(\U{3b3} =f_{y}/f_{x}\), representing uniquely a family of planar curves \(f(x,y)=c\), find, if possible, potentials of the form \(V(x,y)=v(\U{3b3} (x,y))\) which give rise to this family. Such potentials \(V\) will then have as equipotential curves the isoclinic curves \(\U{3b3} =\) constof the family \(\(f(x,y)=c\). We show that, for the problem of admitting a solution, a necessary and sufficient condition must be satisfied by the given \(\U{3b3} (x,y)\). Inferring by reasoning from particular to more general forms, we find analytically a very rich set of slope functions \(\(\U{3b3} (x,y)\) satisfying this condition. In contrast to the (not always solvable) general case \(V=V(x,y)\), in all these cases we can find the potential (v=v(\U{3b3} )\) analytically by quadratures. Several examples of pairs \((\U{3b3} ,v(\U{3b3}))\) are presented.
Authors
Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
George Bozis
University of Thessaloniki, Department of Theoretical Mechanics, Thessaloniki, Greece
Keywords
?
Paper coordinates
George Bozis and Mira-Cristiana Anisiu, A solvable version of the inverse problem of dynamics, Published 1 February 2005 • 2005 IOP Publishing Ltd, , 487-497, , http://10.1088/0266-5611/21/2/005
??
About this paper
Journal
Inverse problem
Publisher Name
IOPscience
DOI
Print ISSN
1361-6420
Online ISSN
0266-5611
google scholar link
[1] Agekyan T A 2003 A basic system of equations in the field of a rotationally symmetric potential Astron. Lett. 29 348-51, Crossref, Google Scholar
[2] Anisiu M-C 2003 PDEs in the inverse problem of dynamics Analysis and Optimization of Differential Systems ed V Barbu et al (Dordrecht: Kluwer Academic) pp 13-20, Crossref, Google Scholar, View article
[3] Anisiu M-C 2004 An alternative point of view on the equations of the inverse problem of dynamics Inverse Problems 20 1865-72, IOPscience, Google Scholar
[4] Borghero F and Bozis G 2002 Isoenergetic families of planar orbits generated by homogeneous potentials Meccanica 37 545-54, Crossref, Google Scholar
[5] Bozis G 1995 The inverse problem of dynamics: basic facts Inverse Problems 11 687-708, IOPscience, Google Scholar
[6] Bozis G 2003 Certain comments on: “Open problems on the eve of the next millennium” by V Szebehely Celest. Mech. Dyn. Astron. 85 219-22, Crossref, Google Scholar
[7] Bozis G and Anisiu M-C 2001 Families of straight lines in planar potentials Rom. Astron. J. 11 27-43, Google Scholar
[8] Bozis G, Anisiu M-C and Blaga C 1997 Inhomogeneous potentials producing homogeneous orbits Astron. Nachr. 318 313-8, Crossref, Google Scholar
[9] Bozis G, Anisiu M-C and Blaga C 2000 A solvable version of the direct problem of dynamics Rom. Astron. J. 10 59-70, Google Scholar
[10] Bozis G and Ichtiaroglou S 1994 Boundary curves for families of planar orbits Celest. Mech. Dyn. Astron. 58 371-85, Crossref, Google Scholar
[11] Contopoulos G and Bozis G 2000 Complex force fields and complex orbits J. Inverse Ill-Posed Probl. 8 147-60, Crossref, Google Scholar
[12] Galiullin A S 1984 Inverse Problems of Dynamics (Moscow: Mir) Google Scholar
[13] Gonzales-Gascon F, Gonzales-Lopez A and Pascual-Broncano P J 1984 On Szebehely’s equation and its connections with Dainelli’s-Whittaker’s equations Celest. Mech. 33 85-97, Crossref, Google Scholar
[14] Goursat E 1945 A Course in Mathematical Analysis, Differential Equations vol II part 2 (New York: Dover), Google Scholar
[15] Hénon M and Heiles C 1964 The applicability of the third integral of motion: some numerical experiments Astron. J. 69 73-9, Crossref, Google Scholar
[16] Puel F 1999 Potentials having two orthogonal families of curves as trajectories Celest. Mech. Dyn. Astron. 74 199-210, Crossref, Google Scholar
[17] Ramirez R and Sadovskaia N 2004 Inverse problems in dynamics Atti Sem. Mat. Fis. Univ. Modena LII at press, Google Scholar
[18] Santilli R M 1978 Foundations of Theoretical Mechanics vol I (New York: Springer), Crossref, Google Scholar
[19] Szebehely V 1974 On the determination of the potential by satellite observations Proc. of the Int. Meeting on Earth’s Rotation by Satellite Observation ed G Proverbio Rend. Sem. Fac. Sc. Univ. Cagliari XLIV (Suppl.) pp 31-5, Google Scholar
[20] Szebehely V 1997 Open problems on the eve of the next millennium Celest. Mech. Dyn. Astron. 65 205-11, Crossref, Google Scholar
[21] Whittaker E T 1961 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge: Cambridge University Press), Google Scholar