The extended Lissajous–Levi-Civita transformation

Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler’s equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous–Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams.     

Concerning the regularizing KS-transformation

In this note we give by means of quaternions in vector notation a new derivation of the KS-transformation acting from a four-dimensional parameter space into the three-dimensional physical space. Using quaternions in vector notation each step in the derivation has an immediate geometrical interpretation. In particular, the KS-transformation appears as the Levi-Civita transformation, formulated in a rotated coordinate system.Dedicated to Professor Otto Volk.     

Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

On CCGG,the De Donder-Weyl Hamiltonian formulation of canonical gauge gravity

This short paper gives a brief overview of the manifestly covariant canonical gauge gravity (CCGG) that is rooted in the De Donder-Weyl Hamiltonian formulation of relativistic field theories, and the proven methodology of the canonical transformation theory. That framework derives, from a few basic physical and mathematical assumptions, equations describing generic matter and gravity dynamics with the spin connection emerging as a Yang Mills-type gauge field. While the interaction of any matter field with spacetime is fixed just by the transformation property of that field, a concrete gravity ansatz is introduced by the choice of the free (kinetic) gravity Hamiltonian. The key elements of this approach are discussed and its implications for particle dynamics and cosmology are presented. New insights: Anomalous Pauli coupling of spinors to curvature and torsion of spacetime, spacetime with (A)dS ground state, inertia, torsion and geometrical vacuum energy, Zero-energy balance of the Universe leading to a vanishing cosmological constant and torsional dark energy.     

Linearization: Laplace vs. Stiefel

The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIII^th century; Laplace seems to be the first to have codified it. We reorganize the classical material around the Theorem of the Moving Frame. Concerning Stiefel's own contribution to the question, on the one hand, we abandon the formalism of Matrix Theory to proceed exclusively in the context of quaternion algebra; on the other hand, we explain how, in the hierarchy of hypercomplex systems, both the KS-transformation and the classical projective decomposition emanate by doubling from the Levi-Civita transformation. We propose three ways of stretching out the projective factoring into four-dimensional coordinate transformations, and offer for each of them a canonical extension into the moment space. One of them is due to Ferrándiz; we prove it to be none other than the extension of Burdet's focal transformation by Liouville's technique. In the course of constructing the other two, we examine the complementarity between two classical methods for transforming Hamiltonian systems, on the one hand, Stiefel's method for raising the dimensions of a system by means of weakly canonical extensions, on the other, Liouville's technique of lowering dimensions through a Reduction induced by ignoration of variables.     

Elimination des noeuds dans le probleme newtonien des quatre corps

Résumé Nous appliquons la méthode des transformations canoniques à variables imposées à la réduction du problème newtonien des quatre corps. L'élimination du centre de gravité étant supposée faite, le problème est ramené à celui des trois corps fictifs. Alors nous menons à bien la réduction dûe aux intégrales des aires explicitement sous forme Hamiltonienne en tenant compte de l'aspect géométrique d'élimination des noeuds préconisé par Jacobi.Nous nous imposons trois fonctions comme nouvelles variables: la troisième intégrale des aires et deux fonctions in variantes; ces deux dernières fonctions resteront nulles lorsque nous prendrons comme troisième axe de coordonnées l'axe défini par le moment cinétique des quatre corps; elles sont choisies en involution avec la troisième intégrale des aires et de crochet un entre elles. Cela nous conduit à déterminer un système de quatorze variables canoniques que nous interprétons géométriquement. Il y a effectivement élimination des moeuds: il s'introduit un pseudo-noeud commun aux deuxième et troisième corps fictifs qui concide avec le noeud du premier corps fictif; ces noeud et pseudo-noeud sont repérés par un paramètre ignorable.

---

Elimination of nodes in the Newtonian four-body problem

We apply the method of canonical trasformations with imposed variables to the reduction of the Newtonian four-body problem. After the elimination of the center of gravity, the problem is reduced to that of three fictitious bodies. Then we proceed to the actual reduction using the integrals of angular momentum, in Hamiltonian formulation, and considering the geometrical aspects of the elimination of the nodes advocated by Jacobi.We impose three functions as new variables: the third integral of angular momentum and two invariant functions; these last two functions will remain null when we take as third coordinate axis the axis, defined by the momentum vector of the four bodies; they are chosen in involution with the third integral of momentum and so that their Poisson bracket is equal to one. Then we determine a system of fourteen canonical variables which have a simple geometrical interpretation. It is an actual elimination of the nodes: a pseudonode for the second and third fictitious bodies is introduced which coincides with the node of the first fictitious body; the node and the pseudo-node are referred to by an ignorable parameter.

     

The planar Hill problem with oblate primary

The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'. This revised version was published online in July 2006 with corrections to the Cover Date.     

Synchronization of perturbed non-linear Hamiltonians

We propose a new method based on Lie transformations for simplifying perturbed Hamiltonians in one degree of freedom. The method is most useful when the unperturbed part has solutions in non-elementary functions. A non-canonical Lie transformation is used to eliminate terms from the perturbation that are not of the same form as those in the main part. The system is thus transformed into a modified version of the principal part. In conjunction with a time transformation, the procedure synchronizes the motions of the perturbed system onto those of the unperturbed part.A specific algorithm is given for systems whose principal part consists of a kinetic energy plus an arbitrary potential which is polynomial in the coordinate; the perturbation applied to the principal part is a polynomial in the coordinate and possibly the momentum.We demonstrate the strategy by applying it in detail to a perturbed Duffing system. Our procedure allow us to avoid treating the system as a perturbed harmonic oscillator. In contrast to a canonical simplification, our method involves only polynomial manipulations in two variables. Only after the change of time do we start manipulating elliptic functions in an exhaustive discussion of the flows.     

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 mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets; the inverse transformation can be built the same way.     

Elimination of the nodes in problems ofn bodies

In application of the Reduction Theorem to the general problem ofn (>-3) bodies, a Mathieu canonical transformation is proposed whereby the new variables separate naturally into (i) a coordinate system on any reduced manifold of constant angular momentum, and (ii) a quadruple made of a pair of ignorable longitudes together with their conjugate momenta. The reduction is built from a binary tree of kinetic frames Explicit transformation formulas are obtained by induction from the top of the tree down to its root at the invariable frame; they are based on the unit quaternions which represent the finite rotations mapping one vector base onto another in the chain of kinetic frames. The development scheme lends itself to automatic processing by computer in a functional language.     

Increased accuracy of computations in the main satellite problem through linearization methods

The set of canonical redundant variables previously introduced by the first author is derived from Cartesian coordinates in a simplified form which allows the reduction of the Kepler problem to four harmonic oscillators with unit frequency. The coordinates are defined to be the direction cosines of the position of the particle along with the inverse of its distance. True anomaly is the new independent variable. The behavior of this new transformation is studied when applied to the numerical integrations of the main problem in satellite theory. In particular, computation time and accuracy of orbits in the new variables are compared with those in K-S and Cartesian variables. It is noteworthy that for high eccentricities the new variables require the least computation time for comparable accuracy, regardless of the integration scheme.     

Canonical Floquet theory

Classical Floquet theory is reviewed with careful attention to the case of repeated eigenvalues common in Hamiltonian systems. Floquet theory generates a canonical transformation to modal variables if the periodic matrix can be made symplectic at the initial time. It is shown that this symplectic normalization can always be carried out, again with careful attention to the degenerate case. The periodic modal vectors and canonical modal variables can always be chosen to be purely real. It is possible to introduce real valued action-angle variables for all modes. Physical interpretation of the canonical degenerate normal modal variables are offered. Finally, it is shown that this transformation enables canonical perturbation theory to be carried out using Floquet modal variables.     

A note concerning the TR-transformation

The canonical transformation which Scheifele (1970) proposes to make a coordinate of the true anomaly is the product of a Whittaker transformation by an extension to space-time of the one-parameter family of canonical transformations that Hill (1913) defined for the same purpose.     

The KS-transformation in hypercomplex form

In this note the KS-transformation introduced by Kustaanheimo and Stiefel into Celestial Mechanics is formulated in terms of hypercomplex numbers as the product of a quaternion and its antiinvolute. Therefore it represents a particular morphism of the real algebra of quaternions-having for image a three-dimensional real linear subspace-and also anatural generalization of the Levi-Civita transformation. The quaternion matrix of the product leads to the KS-matrix; the bilinear relation and the two identities which play a central role in the KS-theory are easily derived. A suitable quaternion gauge-transformation is given which leads to the well-known fibration of the four-dimensional space. In addition several geometrical interpretations are brought out.     

Perturbed Gylden Systems and Time-Dependent Delaunay-Like Transformations

We study the possibilities and limitations of the application of generalized Delaunay-like transformations (in the 6-dimensional phase space) and TR-like mappings (in the 8-dimensional, extended phase space) to perturbed two-body problems with a time-varying Keplerian parameter μ(t), that is, to Gylden-type systems. For the sake of theoretical completeness, both negative- and positive-energy motion (with nonstationary coupling parameter) are, in principle, considered. Our developments are intended to introduce canonical variables parallelling the classical ones of Delaunay and the Delaunay-Similar variables of Scheifele. This revised version was published online in August 2006 with corrections to the Cover Date.     

Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semianalytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.     

New canonical variables for orbital and rotational motions

A new canonical transformation of freedom two was found. By using this, we derived three new sets of canonical variables for the orbital motion and two for the rotational motion. New canonical variables have clear physical meanings and remain well-defined in the case when the classical sets become ill-defined, for example, when the eccentricity and/or the inclination is small for the elliptic orbital motion.     

A new algorithm for the Lie transformation

Following Hori, the Lie transformation is presented in a form that is independent of any extraneous parameters. The transformation is canonical, and its inverse is obtained by changing the sign of the generating function. The introduction of a small parameter into the generating function and the Hamiltonian then yields a recursive, triangular algorithm. The case of a Hamiltonian containing the time explicitly is included by adjoining an additional pair of conjugate variables. The necessary and sufficient condition that this transformation be identical to Deprit's transformation is given as a recursive relation between successive terms in the generating functions. Explicit formulas are obtained through the sixth order.After submitting the present paper the author learned of similar and independent work by Campbell and Jefferys and by Kamel (Ph.D. thesis).     

The generalized canonical form of Hori's method for non-canonical systems

Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables or momenta and solved by perturbation theory based on Lie series. The application of Hori's method for generalized canonical system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system is no longer necessary. In this paper, we wish to show some peculiarities of this technique.