Abstract
We consider the following version of the inverse problem of Dynamics: given a monoparametric family of planar curves, find the force field, conservative or not, which determines a material point to move on the curves of that family. We present the partial differential equations which are satisfied by the potential and we clarify the role of the energy function. Due to the nonuniqueness of the solution of the PDEs, it is natural to look for force fields in certain classes of functions (e.g., polynomial, homogeneous, or satisfying also another PDE). In connection with the inverse problem of Dynamics, programmed motion is studied imposing the supplementary condition that the orbits lie in a preassigned region of the plane. Applications in Celestial Mechanics, Geometrical Optics and Fluid Dynamics are given.
Authors
Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Keywords
The inverse problem of Dynamics.
Paper coordinates
M.-C. Anisiu, The Planar Inverse Problem of Dynamics, Mathematics Without Boundaries, Surveys in Interdisciplinary Researchhttps://doi.org/10.1007/978-1-4939-1124-0_1
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Mathematics Without Boundaries
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[1] Anisiu, M.-C., Pál, Á.: Special families of orbits for the Hénon-Heiles type potential. Rom. Astron. J. 9, 179–185 (1999) Google Scholar
[2] Anisiu, M.-C.: The Equations of the Inverse Problem of Dynamics. House of the Book of Science, Cluj-Napoca (2003) (in Romanian) Google Scholar
[3] Anisiu, M.-C.: PDEs in the inverse problem of Dynamics. In: Barbu, V., et al. (eds.) Analysis and Optimization of Differential Systems, pp. 13–20. Kluwer Academic, Boston (2003) Chapter Google Scholar
[4] Anisiu, M.-C.: An alternative point of view on the equations of the inverse problem of dynamics. Inverse Probl. 20, 1865–1872 (2004), Article MathSciNet MATH Google Scholar
[5] Anisiu, M.-C., Bozis, G.: Programmed motion for a class of families of planar orbits. Inverse Probl. 16, 19–32 (2000), Article MathSciNet MATH Google Scholar
[6] Antonov, V.A., Timoshkova, E.I.: Simple trajectories in a rotationally symmetric gravitational field. Astron. Rep. 37, 138–144 (1993) MathSciNet Google Scholar
[7] Boccaletti, D., Pucacco, G.: Theory of Orbits I. Springer, Berlin/Heidelberg (1996) Book Google Scholar
[8] Borghero, F., Bozis, G.: Isoenergetic families of planar orbits generated by homogeneous potentials. Meccanica 37, 545–554 (2002) Article MathSciNet MATH Google Scholar
[9] Borghero, F., Bozis, G.: A two-dimensional inverse problem of geometrical optics. J. Phys. A Math. Gen. 38, 175–184 (2005) Article MathSciNet MATH Google Scholar
[10] Bozis, G.: Inverse problem with two parametric families of planar orbits. Celest. Mech. Dyn. Astron. 60, 161–172 (1994) Article MathSciNet MATH Google Scholar
[11] Bozis, G.: Szebehely inverse problem for finite symmetrical material concentrations. Astron. Astrophys. 134, 360–364 (1984) MathSciNet MATH Google Scholar
[12] Bozis, G.: Family boundary curves for autonomous dynamical systems. Celest. Mech. 31, 129–142 (1983)Article MathSciNet MATH Google Scholar
[13] Bozis, G.: The inverse problem of dynamics: basic facts. Inverse Probl. 11, 687–708 (1995) Article MathSciNet MATH Google Scholar
[14] Bozis, G., Anisiu, M.-C.: Families of straight lines in planar potentials. Rom. Astron. J. 11, 27–43 (2001) Google Scholar
[15] Bozis, G., Anisiu, M.-C.: A solvable version of the inverse problem of dynamics. Inverse Probl. 21, 487–497 (2005) Article MathSciNet MATH Google Scholar
[16] Bozis, G., Anisiu, M.-C.: Programmed motion in the presence of homogeneity. Astron. Nachr. 330, 791–796 (2009) Article MATH Google Scholar
[17] Bozis, G., Anisiu, M.-C., Blaga, C.: Inhomogeneous potentials producing homogeneous orbits. Astron. Nachr. 318, 313–318 (1997) Article MATH Google Scholar
[18] Bozis, G., Borghero, F.: An inverse problem in fluid dynamics. In: Monaco, R., et al. (eds.) Waves and Stability in Continuous Media – WASCOM 2001, pp. 89–94. World Scientific Publishing, Singapore (2002) Google Scholar
[19] Bozis, G., Grigoriadou, S.: Families of planar orbits generated by homogeneous potentials. Celest. Mech. Dyn. Astron. 57, 461–472 (1993) Article MathSciNet MATH Google Scholar
[20] Bozis, G., Ichtiaroglou, S.: Boundary curves for families of planar orbits. Celest. Mech. Dyn. Astron. 58, 371–385 (1994) Article MathSciNet Google Scholar
[21] Broucke, R., Lass, H.: On Szebehely’s equation for the potential of a prescribed family of orbits. Celest. Mech. 16, 215–225 (1977) Article MathSciNet MATH Google Scholar
[22] Caranicolas, N.D.: Potentials for the central parts of a barred galaxy. Astron. Astrophys. 332, 88–92 (1998) Google Scholar
[23] Caranicolas, N.D., Innanen, K.A.: Periodic motion in perturbed elliptic oscillators. Astron. J. 103, 1308–1312 (1992) Article Google Scholar
[24] Carrasco, D., Vidal, C.: Periodic solutions, stability and non-integrability in a generalized Hénon-Heiles Hamiltonian system. J. Nonlinear Math. Phys. 20, 199–213 (2013) Article MathSciNet Google Scholar
[25] Contopoulos, G., Bozis, G.: Complex force fields and complex orbits. J. Inverse Ill-Posed Probl. 8, 1–14 (2000) Article MathSciNet Google Scholar
[26] Contopoulos, G., Zikides, M.: Periodic orbits and ergodic components of a resonant dynamical system. Astron. Astrophys. 90, 198–203 (1980) MathSciNet Google Scholar
[27] Dainelli, U.: Sul movimento per una linea qualunque. Giorn. Mat. 18, 271–300 (1880) MATH Google Scholar
[28] Érdi, B., Bozis, G.: On the adelphic potentials compatible with a set of planar orbits. Celest. Mech. Dyn. Astron. 60, 421–430 (1994) Article MATH Google Scholar
[29] Galiullin, A.S.: Inverse Problems. Mir, Moscow (1984) Google Scholar
[30] Gonzáles-Gascón, F., Gonzáles-Lopéz, A., Pascual-Broncano, P.J.: On Szebehely’s equation and its connection with Dainelli’s-Whittaker’s equations. Celest. Mech. 33, 85–97 (1984) Article Google Scholar
[31] Grigoriadou, S.: The inverse problem of dynamics and Darboux’s integrability criterion. Inverse Probl. 15, 1621–1637 (1999) Article MathSciNet MATH Google Scholar
[32] Hénon, M., Heiles, C.: The applicability of the third integral of motion, some numerical experiments. Astron. J. 69, 73–79 (1964) Article Google Scholar
[33] Howard, J.E., Meiss, J.D.: Straight line orbits in Hamiltonian flows. Celest. Mech. Dyn. Astron. 105, 337–352 (2009) Article MathSciNet MATH Google Scholar
[34] Ichtiaroglou, S., Meletlidou, E.: On monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants. J. Phys. A: Math. Gen. 23, 3673–3679 (1990) Article MathSciNet MATH Google Scholar
[35] Kostov, N.A., Gerdjikov, V.S., Mioc, V.: Exact solutions for a class of integrable Hénon-Heiles-type systems. J. Math. Phys. 51, 022702.1–022702.13 (2010) Article MathSciNet Google Scholar
[36] Luneburg, R.K.: Mathematical Theory of Optics. University of California Press, Berkeley/Los Angeles (1964) Google Scholar
[37] van der Merwe, P.du T.: Solvable forms of a generalized Hénon-Heiles system. Phys. Lett. A 156, 216–220 (1991) Google Scholar
[38] Miller, R.H., Smith, B.F.: Dynamics of a stellar bar. Astrophys. J. 227, 785–797 (1979) Article Google Scholar
[39] Mioc, V., Paşca, D., Stoica, C.: Collision and escape orbits in a generalized Hénon-Heiles model. Nonlinear Anal. Real 11, 920–931 (2010) Article MATH Google Scholar
[40] Molnár, S.: Applications of Szebehely’s equation. Celest. Mech. 29, 81–88 (1981) Article Google Scholar
[41] Pál, Á., Anisiu, M.-C.: On the two-dimensional inverse problem of dynamics. Astron. Nachr. 317, 205–209 (1996) Article MATH Google Scholar
[42] Puel, F.: Formulation intrinseque de l’équation de Szebehely. Celest. Mech. 32, 209–212 (1984) Article MATH Google Scholar
[43] Serrin, J.: Mathematical Principles of Classical Fluid Mechanics. In: Flugge, S. (ed.) Fluid Dynamics I. Encyclopaedia of Physics, vol. 8/1, pp. 125–350, Springer, Berlin/Heidelberg (1959) Google Scholar
[44] Szebehely, V.: On the determination of the potential by satellite observations. In: Proverbio, G. (ed.) Proceedings of the International Meeting on Earth’s Rotation by Satellite Observation, pp. 31–35. The University of Cagliari, Bologna (1974) Google Scholar
[45] Szebehely, V., Lundberg, J., McGahee, W.J.: Potential in the central bar structure. Astrophys. J. 239, 880–881 (1980) Article MathSciNet Google Scholar
[46] Whittaker, E.T.: Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1904) MATH Google Scholar
[47] Zotos, E.E.: Using new dynamical indicators to distinguish between order and chaos in a galactic potential producing exact periodic orbits and chaotic components. Astron. Astrophys. Trans. 4, 635–654 (2012) Google Scholar