The Lissajous transformation II. Normalization

  title={The Lissajous transformation II. Normalization},
  author={A. Deprit and Antonio Elipe},
  journal={Celestial Mechanics and Dynamical Astronomy},
  • A. Deprit, A. Elipe
  • Published 1 September 1991
  • Physics, Mathematics
  • Celestial Mechanics and Dynamical Astronomy
Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the… 
Bifurcations in biparametric quadratic potentials. II.
The occurrence of bifurcations is analyzed and the biforcation lines in the parameter plane for one of these classes are obtained, thus complementing the work done in a previous paper where the other class was analyzed.
Some Analytical Results in Dynamics of Spheroidal Galaxies
The surfaces of section in a harmonic oscillator potential, perturbed by quartic terms, are obtained analytically. A succession of action‐angle, Lissajous and Lie transformations near the 1:1
The Hénon and Heiles Problem in Three Dimensions.
The second part of a research on the Henon and Heiles system in three dimensions is presented. We focus on motions around the origin, where the system may be treated as a case of three perturbed
Quadratic Hamiltonians on pinched spheres
Among other dynamical systems, two normalized perturbed elliptic oscillators in resonance 1:1 may be represented by a parametric quadratic form in a set of Cartesian variables whose Poisson structure
Bifurcation Sequences in the Symmetric 1: 1 Hamiltonian Resonance
We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant
Normalization of Resonant Hamiltonians
The definition of integrability is simple to state; an autonomous N degree of freedom Hamiltonian is integrable if N independent global invariants exist and these are in involution with each other.1
Normalization and the detection of integrability: The generalized Van Der Waals potential
Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled


Some properties of an intrinsically degenerate Hamiltonian system
Using theoretical arguments we study some properties of a time independent, two-dimensional Hamiltonian system in the 1:1 resonance case. In particular we find the frequency of the oscillation near
The Lissajous transformation III. Parametric bifurcations
We examine a parametric family of cubic perturbed 1-1 resonant harmonic oscillators with an aim to understanding the phase flows of the reduced system. Variation of the parameters leads the system
Canonical transformations depending on a small parameter
The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonical
Comparison of classical and quantal spectra for the Henon-Heiles potential
The quantal energy spectrum is compared with the classical motion for a modified Henon-Heiles potential. There is good agreement between the amount of classical irregular motion and the proportion of
Integrability of Hamiltonians with third‐ and fourth‐degree polynomial potentials
The weak‐Painleve property, as a criterion of integrability, is applied to the case of simple Hamiltonians describing the motion of a particle in two‐dimensional polynomial potentials of degree three
Integrable Hamiltonian Systems and the Painleve Property
A direct method is described for obtaining conditions under which certain $N$-degree-of-freedom Hamiltonian systems are integrable, i.e., possess $N$ integrals in involution. This method consists of
Coupled quartic anharmonic oscillators, Painlevé analysis, and integrability.
A detailed and systematic investigation of the Painleve (P) properties of coupled quartic anharmonic oscillators shows that there exist four different parametric cases possessing P properties, two are identified with strong P property and the other two with weak P property.
Periodic orbits in a resonant dynamical system
We study the periodic orbits in a two dimensional dynamical system, symmetric with respect to both axes, with two equal or nearly equal frequencies. It is shown that the periodic orbits can be found