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Register a new userThe three-body problem and the Hannay angle
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Added by
Alessandro D. A. M. Spallicci
Publication date
2003
fields
Physics
and research's language is
English
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The Hannay angle has been previously studied for a celestial circular restricted three-body system by means of an adiabatic approach. In the present work, three main results are obtained. Firstly, a formal connection between perturbation theory and the Hamiltonian adiabatic approach shows that both lead to the Hannay angle; it is thus emphasised that this effect is already contained in classical celestial mechanics, although not yet defined nor evaluated separately. Secondly, a more general expression of the Hannay angle, valid for an action-dependent potential is given; such a generalised expression takes into account that the restricted three-body problem is a time-dependent, two degrees of freedom problem even when restricted to the circular motion of the test body. Consequently, (some of) the eccentricity terms cannot be neglected {it a priori}. Thirdly, we present a new numerical estimate for the Earth adiabatically driven by Jupiter. We also point out errors in a previous derivation of the Hannay angle for the circular restricted three-body problem, with an action-independent potential.
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The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation of a mechanical problem to fewer degrees of freedom, a process known as dynamical reduction. However, extant reductions are either non-general, or hide the problems symmetry or include unexplained definitions. This paper presents a dynamical reduction that avoids these issues, and hence is general and natural. Any three-body configuration defines a triangle, and its orientation in space. Accordingly, we decompose the dynamical variables into the geometry (shape + size) and orientation of the triangle. The geometry variables are shown to describe the motion of an abstract point in a curved 3d space, subject to a potential-derived force and a magnetic-like force with a monopole charge. The orientation variables are shown to obey a dynamics analogous to the Euler equations for a rotating rigid body, only here the moments of inertia depend on the geometry variables, rather than being constant. The reduction rests on a novel symmetric solution to the center of mass constraint inspired by Lagranges solution to the cubic. The formulation of the orientation variables is novel and rests on a little known generalization of the Euler-Lagrange equations to non-coordinate velocities. Applications to special exact solutions and to the statistical solution are described or discussed. Moreover, a generalization to the four-body problem is presented.
Eulers three-body problem is the problem of solving for the motion of a particle moving in a Newtonian potential generated by two point sources fixed in space. This system is integrable in the Liouville sense. We consider the Euler problem with the inverse-square potential, which can be seen as a natural generalization of the three-body problem to higher-dimensional Newtonian theory. We identify a family of stable stationary orbits in the generalized Euler problem. These orbits guarantee the existence of stable bound orbits. Applying the Poincare map method to these orbits, we show that stable bound chaotic orbits appear. As a result, we conclude that the generalized Euler problem is nonintegrable.
We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the Brutus N-body code to include Post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the Brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass <= 10 MSun, the system behavior is indistinguishable from the Newtonian case, and the resonance always ends in a binary and one escaping body. For a mass scaling up to 1e7 MSun, relativity gradually becomes more prominent, but the majority of the systems still dissolve. The first mergers start to appear for a mass of ~1e5 MSun, and between 1e7 MSun and 1e9 MSun all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling >= 1e9 MSun, all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of 1e7 MSun potentially leaves an imprint in the mass function of supermassive black holes.
We obtain the adiabatic Berry phase by defining a generalised gauge potential whose line integral gives the phase holonomy for arbitrary evolutions of parameters. Keeping in mind that for classical integrable systems it is hardly clear how to obtain open-path Hannay angle, we establish a connection between the open-path Berry phase and Hannay angle by using the parametrised coherent state approach. Using the semiclassical wavefunction we analyse the open-path Berry phase and obtain the open-path Hannay angle. Further, by expressing the adiabatic Berry phase in terms of the commutator of instantaneous projectors with its differential and using Wigner representation of operators we obtain the Poisson bracket between distribution function and its differential. This enables us to talk about the classical limit of the phase holonomy which yields the angle holonomy for open-paths. An operational definition of Hannay angle is provided based on the idea of classical limit of quantum mechanical inner product. A probable application of the open-path Berry phase and Hannay angle to wave-packet revival phenomena is also pointed out.
Based on a three-potential formalism we propose mathematically well-behaved Faddeev-type integral equations for the atomic three-body problem and descibe their solutions in Coulomb-Sturmian space representation. Although the system contains only long-range Coulomb interactions these equations allow us to reach solution by approximating only some auxiliary short-range type potentials. We outline the method for bound states and demonstrate its power in benchmark calculations. We can report a fast convergence in angular momentum channels.
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