Properties of Orbits Space of Indefinite Hamiltonian Resonances

Authors
Reza Mazrooei-Sebdani
Omid Toghraei
Isfahan University of Technology
Abstract
ABSTRACT

The study of Hamiltonian systems around the elliptic equilibrium points, which is a non-generic subject in the study of Hamiltonian systems, has received attention in recent decades. Such systems appear in many applied models, including molecular physics, galactic dynamics, and mechanics.

Consider a Hamiltonian of n degrees of freedom, whose quadratic part is as follows,



(1)
Η2q.p=12j=1nωjqj2+pj2, ωjϵZ, j=1.….n.



The vector ω≔ω1,ω2, …, ωn is called frequency vector (related to Η2 ) and its components are called frequency. If all frequencies are non-zero, ω is called non-degenerate, and if at least one of the frequencies is zero, we say ω is degenerate.

Definition. Any integer-valued vector perpendicular to the frequency vector ω is called a resonance or annihilator vector. In other words, k∈Zn0 is a resonant vector for the frequency vector ω if

k,ω=j=1nkjωj=0.

If there is such an annihilator vector for ω , we call the ω resonance frequency vector. If at least one of the components of the non-degenerate resonance frequency vector ω is negative, the Hamiltonian resonance is called indefinite.

Material and methods

In this paper, we focus on indefinite Hamiltonian resonances, so that by introducing some important physical models, we review their occurrence in different parts of science including the satellite problem, plasma interactions, Pais-Ollenbeck oscillators (PU) and a problem of cosmology. Then, by using the normal form of indefinite Hamiltonian resonances, we will discuss the space structure of the corresponding vector field.

Results and discussion

As we mentioned there are many models passing indefinite Hamiltonian resonances. We clear the space of orbits of indefinite Hamiltonian resonances topologically. Specifically, we will see it can be unbounded in the comparison to space of orbits of other Hamiltonian resonances.
Keywords
Hamiltonian resonance
Indefinite resonance
Normal Form
Action-angle coordinates
Space of orbits
V. I. Arnol’d, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; in Dynamical Systems III (ed. V.I. Arnol’d). Encyc. Math. Sciences, Springer (2006).



K. Andrzejewski, J. Gonera and P. Maslanka, Euclidean Path Integral and Higher-Derivative Theories, Prog. Theor. Phys. 125 (2011) 247-260.



K. Bolonek and P. Kosinski, Hamiltonian structures for Pais-Uhlenbeck oscillator, Acta Phys. Pol. B 36 (2005) 2115-2131.



V. V. Beletskii, Motion of an artificial satellite about its center of mass, in Artificial Earth Satellites, 1, pp. 25--43; Academy of Sciences, Moscow, 1958 (in Russian); Artificial Earth Satellites, Vols. 1 and 2, L.V. Kurnosova, ed., pp. 30--54, Plenum Press, New York, (1960).



D. Carrasco, J. F. Palaci’an, C. Vidal, J. Vidarte and P. Yanguas, Dynamics of Axially Symmetric Perturbed Hamiltonians in 1 : 1 : 1 Resonance”, J. Nonlinear Sci., 28 (2018) 1293-1359.



D. Carcamo-Diaz, J. F. Palacian, C. Vidal and P. Yanguas, Nonlinear Stability in the Spatial Attitude Motion of a Satellite in a Circular Orbit, SIAM J. Applied Dynamical Systems, 20 (2021) 1421-1463.



D.B. DeBra and R.H. Delp, Rigid body attitude stability and natural frequencies in a circular orbit, J. Astronaut. Sci., 8 (1961) 14-17.



R.H. Delp, Attitude Motion of a Small Satellite in an Inverse-Square Central-Force Field, Technical report, Lockheed Missiles and Space Division, (1958), 417670.



S. Ferrer, H. Hanßmann, J. Palacin and P. Yanguas, Hamiltonian oscillators in 1:1:1 resonance: normalization and integrability, J. Nonl. Sci. 10 (2000) 145–174.



S. Ferrer, J. Palacin and P. Yanguas, On perturbed oscillators in 1:1:1 resonance: the case of axially symmetric cubic potentials. J. Geom. Phys. 40 (2002) 320–369.



A. Friedmann, Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes, Z. Phys. A 21 (1922) 326–332.



A. Friedmann, Über die Krümmung des Raumes, Z. Phys. A 10 (1922) 377–386.



H. Hanßmann, R. Mazrooei-Sebdani and F. Verhulst, The 1 : 2 : 4 resonance in a particle chain, Indagat. Math. 32 (2021) 101–120.



H. Hanßmann and J. C. Van der Meer, Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems, J. Dyn. Differ. Equ., 17 (2005) 455–474.



Hoveijn I., Aspects of resonance in dynamical systems, Ph.D. Thesis (1992).



I. Hoveijn and F. Verhulst, Chaos in the 1 : 2 : 3 Hamiltonian normal form. Physica D, 44 (1990) 397–406.



P.C. Hughes, Spacecraft Attitude Dynamics, Dover, Mineola, New York, 2004.



C. S. Kueny and P. J. Morrison, Nonlinear instability and chaos in plasma wave–wave interactions. II. Numerical methods and results, Physics of Plasmas, Vol 2, No. 6, (1995).



G. Lemaître, Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques, Ann. Soc. Sci. Brux. A 47 (1927) 49–59.



G. Lemaître, L’univers en expansion, Ann. Soc. Sci. Brux. A 53 (1933) 51–85.



A. P. Markeev, Linear Hamiltonian Systems and Certain Problems of Stability of Motion of a Satellite in Relation to its Center of Mass, NITs Regular and Chaotic Dynamics, Institute of Computational Analysis, Moscow-Izhevsk, Moscow, (2009) (in Russian).



A. P. Markeev and A. G. Sokolskii, On the stability of relative equilibrium of a satellite in a circular orbit, Kosmic. Issled., 13 (1975), pp. 139--146 (in Russian); Cosm. Res., 13 (1975) 119-125.



S. R. Marandi and V. J. Modi, Attitude stability of rigid satellites via a normalized Hamiltonian, Acta Astronaut., 19 (1989) 287-299.



L. Martinet, P. Magnenat and F. Verhulst, On the number of isolating integrals in resonant systems with 3 degrees of freedom, Celestial mechanics. 25 (1981) 93–99.



R. Mazrooei-Sebdani and E. Hakimi, Periodic Klein-Gordon chains with three particles in 1 : 2 : 2 resonance, J. Dyn. Differ. Equ. 34 (2022) 1349-1370.



R. Mazrooei-Sebdani and E. Hakimi, All relative equilibria of Hamiltonian in detuned 1 : 2 : 3 resonance, J. Differ. Equ. 292 (2021) 501-533.



R. Mazrooei-Sebdani and E. Hakimi, Non-degenerate Hamiltonian Hopf bifurcations in ω : 3 : 6 resonance (ω = 1 or 2), Regul. Chaotic Dyn. 25 (2020) 522-536.



R. Mazrooei-Sebdani and E. Hakimi, On detuned 1 : 1 : 3 Hamiltonian resonance with cases of symmetric cubic and quartic potentials, Chaos. 30 (2020) 093119.



R. Mazrooei-Sebdani and Z. Yousefi, Lagrangian Fibrations in Coupled Resonant Oscillators in the Paired Case of Four Degrees of Freedom Containing Swinging Spring Oscillator, Proc. Math. Phys. Eng. Sci. 478 (2022) 1-20.



K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Third Edition Applied Mathematical Sciences 90, Springer, New York, (2017).



M. Ostrogradsky, Memoires sur les equations differentielles relatives au probleme des isoperim ` etres, Mem. Acad. St. ` Petersbourg VI 4, 385517 (1850).



A. Pais, G.E. Uhlenbeck, On field theories with non-localized action Phys. Rev. 79 (1950) 145.



J. T. B. Oude Groeniger, The 1 : 3 : 4 resonance, Dynamics near a resonant equilibrium M.Sc. thesis, Utrecht University, (2018).



J. F. Palacián, C. Vidal, J. Vidarte and P. Yanguas, Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential, Nonlinearity 32 (2019) 3406–3444.



H. P. Robertson, Kinematics and world structure Astrophys. J. 82 (1935) 284–301.



H. P. Robertson, Kinematics and world structure II Astrophys. J. 83 (1936) 187–201.



H. P. Robertson,1936 Kinematics and world structure III Astrophys. J. 83 (1936) 257–71.



J. A. Sanders, F. Verhulst, J. Murdock, Averaging methods in nonlinear dynamical systems, 2nd edn. Applied Mathematical Sciences, vol. 59. Berlin, Germany: Springer (2007).



A. V. Smilga, Comments on the Dynamics of the Pais–Uhlenbeck Oscillator, SIGMA 5 (2009) 017, arXiv:0808.0139.



E. Van der Aa., First order resonances in three-degrees-of-freedom systems, Celest. Mech. 31 (1983) 163–191.



E. Van der Aa, and J. A. Sanders, On the 1:2:1- resonance, its periodic orbits and integrals, in Asymptotic Analysis, from Theory to Application, ed. F. Verhulst, Lecture Notes in Mathematics 711, Springer-Verlag, Berlin (1979).



E. Van der Aa and M. De. Winkel, Hamiltonian systems in 1 : 2 : omega–resonance (omega = 5 or 6), Int. J. Nonlin. Mech., 29 (1994) 261–270.



E. Van der Aa and F. Verhulst, Asymptotic integrability and periodic solutions of a Hamiltonian system in 1 : 2 : 2-resonance, SIAM J. Math. Anal. 15 (1984) 890–911.



A. G. Walker, On Milne’s theory of world-structure Proc. Lond. Math. Soc. 42 (1937) 90–127.



S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Texts in Applied Mathematics, vol. 2. Berlin, Germany: Springer (2003).



R. P. Woodard, Avoiding Dark Energy with 1/R Modifications of Gravity, Lect. Notes Phys. 720 (2007) 403-433.



V. E. Zhavnerchik, On the stability of autonomous systems in the presence of several resonances, Prikl. Mat. Mekh., 43 (1979), pp. 229--234 (in Russian); J. Appl. Math. Mech., 43 (1979) 247-253.



V. E. Zhavnerchik, On stability in the presence of multiple resonance of odd order, Prikl. Mat. Mekh., 44 (1980), pp. 971--976 (in Russian); J. Appl. Math. Mech., 44 (1980) 692-696.