On orbits for a particular case of axial symmetry (original) (raw)

On the determination of the potential function from given orbits

Czechoslovak Mathematical Journal, 2008

The paper deals with the problem of finding the field of force that generates a given (N − 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov's problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand's problem. At the end we establish the relation between Bertrand's problem and the solutions to the Heun differential equation. We illustrate our results by several examples.

Periodic orbit theory revisited in the anisotropic Kepler problem

Progress of Theoretical and Experimental Physics

Gutzwiller's trace formula for the anisotropic Kepler problem (AKP) is Fourier transformed with a convenient variable u = 1/ √ −2E, which takes care of the scaling property of the AKP action S(E). The proper symmetrization procedure (Gutzwiller's prescription) is used by the introduction of half orbits that close under symmetry transformations, so that the 2D semiclassical formulas correctly match the quantum subsectors m π = 0 + and m π = 0 −. Response functions constructed from half orbits in the periodic orbit theory (POT) side are explicitly given. In particular, the response function g X from the X-symmetric half orbit has an amplitude where the root of the monodromy determinant is inverse hyperbolic. The resultant weighted densities of periodic orbits D m=0 e (φ) and D m=0 o (φ) from both quantum subsectors show peaks at the actions of the periodic orbits with correct peak heights and widths corresponding to their Lyapunov exponents. The formulation takes care of the cutoff of the energy levels, and the agreement between the D(φ)s of the quantum mechanical (QM) and POT sides is observed to be independent of the choice of cutoff. The systematics appearing in the densities of the periodic orbits is explained in terms of features of the periodic orbits. It is shown that, from quantum energy levels, one can extract information on AKP periodic orbits, even the Lyapunov exponents-the success of inverse quantum chaology in AKP.

Monoparametric Families of Orbits Produced by Planar Potentials

Axioms

We study the motion of a test particle on the xy−plane. The particle trajectories are given by a one-parameter family of orbits f(x,y) = c, where c = const. By using the tools of the 2D inverse problem of Newtonian dynamics, we find two-dimensional potentials that produce a pre-assigned monoparametric family of regular orbits f(x,y)=c that can be represented by the “slope function” γ=fyfx uniquely. We apply a new methodology in order to find potentials depending on specific arguments, i.e., potentials of the form V(x,y)=P(u) where u=x2+y2,xy,x3−y3,xy (x,y≠ 0). Then, we establish one differential condition for the family of orbits f(x,y) = c. If it is satisfied, it guarantees the existence of such a potential, generating the above family of planar orbits. Then, the potential function V=V(x,y) is found by quadratures. For known families of curves, e.g., ellipse, the logarithmic spiral, the lemniscate of Bernoulli, and circles, we find homogeneous and polynomial potentials that are com...

On the figure eigth orbit of the three-body problem

Recibido el 6 de marzo de 2003; aceptado el 31 de marzo de 2003 A new solution to the three-body problem interacting through gravitational forces with equal masses and zero angular momentum, has been recently discovered. This is a periodic symmetric orbit where the particles follow a figure eight trajectory in the plane. They alternate between six isosceles-aligned positions and six isosceles triangle positions in a periodic orbit composed by twelve equivalent segments. The condition of zero angular momentum is considered assuming that the three masses can be equal or different, yielding in both cases the same final expression for the kinetic energy. We found that the property of this orbit of having isosceles configurations, is a general feature to be found in any orbit of the equal-mass case, associated with an increase of π/6 in one angle of our set of coordinates. The figure-eight solution is determined by expanding two of our coordinates in a Fourier series of that angle, by using the Jacobi-Maupertuis principle as opposed to the standard Lagrangian action. The time and the angle conjugated to the angular momentum are also expressed in terms of that same angle.

The Planar Three-Body Problem, Symmetries and Periodic Orbits

Qualitative Theory of Dynamical Systems, 2009

We study the planar three-body problem using the principal axes of inertia frame, determined by one Euler angle in the planar case. Three variables are needed in this frame: two distances R 1 and R 2 , related with the principal moments of inertia on the plane of motion of the particles, and an auxiliary angle σ. We also connect these coordinates with the shape sphere of the similarity class of triangles and we give a geometric interpretation of the angle σ. We then write the Hamiltonian and equations of motion in these coordinates to find the symmetries of involution and symmetric periodic orbits. To exemplify this method, we calculate periodic orbits of a three-body problem with two or three equal masses.

The Orbit Method in Geometry and Physics

2003

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Symmetrical periodic orbits in the three body problem - the variational approach

The variational method for searching for plane symmetrical periodic orbits is applied to the three body problem. The orbits are defined as the minimizers of the Lagrangian action functional. According to Barutello et al. (2004), all finite symmetry groups can be reduced to a short list. A number of symmetrical orbits are obtained, which are equivariant with respect to these groups with several values of the rotating frame angular velocity omega. The properties of these symmetrical orbits are discussed. All orbits are compared with results of numerical simulations. This approach makes it possible to formulate the restricted four body problem in a classical way.

On the orbital periods for a particular case of spherical symmetry

Serbian Astronomical Journal, 2008

A particular case of steady state and spherical symmetry - the so-called logarithmic potential introduced as the first approximation for dark coronae of galaxies - is studied. Both time and angle dependence of the distance to the centre for the orbit of a bound test particle with arbitrary initial conditions are calculated numerically. The main attention is paid to the ratio of the sidereal period to the anomalistic one. It is found that this ratio is only slightly variable for a given mean distance to the centre and to increase with increasing orbital eccentricity. This quantitative result may be explained by the fact that the cumulative mass dependence on the distance corresponding to the logarithmic potential obeys a power law, the case where the ratio of the second derivative of the potential to the square of angular velocity for the same distance is constant. On the other hand, compared to the period of circular motion both periods increase with increasing eccentricity.

Nonlinear realizations, the orbit method and

2016

The orbit method is used to describe the centre of mass motion of the system of particles with fixed charge to mass ratio moving in homogeneous magnetic field and confined by harmonic potential. The nonlinear action of symmetry group on phase space is identified and compared with the one obtained with the help of Eisenhart-Duval lift.

Exact, e = 0, Solutions for General Power-Law Potentials. I. Classical Orbits

1994

For zero energy, E = 0, we derive exact, classical solutions for all power-law potentials, V (r) = −γ/r ν, with γ> 0 and − ∞ < ν < ∞. When the angular momentum is non-zero, these solutions lead to the orbits ρ(t) = [cosµ(ϕ(t) − ϕ0(t))] 1/µ, for all µ ≡ ν/2 − 1 ̸ = 0. When ν> 2, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of ϕ(t) and ϕ0(t), as functions of t, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special ν = 2 case.