This paper deals with the effective computation of normal forms, centre manifolds and first integrals in Hamiltonian mechanics. These calculations are very useful since they allow us, among other things, to give explicit estimates on the diffusion time and to compute invariant tori. The approach presented here is based on the algebraic manipulation of formal series with numerical coefficients for them. This, together with a very efficient software implementation, allows big savings in memory and execution time in comparison with the use of commercial algebraic manipulators. The algorithms are discussed together with their C/C++ implementations, and they are applied to some concrete examples from celestial mechanics.

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A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems

We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using $N$ points, a Newton step requires $O(N)$ storage and $O(N\log N)$ operations in Fourier discretization or $O(N)$ operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem ...

Computation of limit cycles and their isochrons: Fast algorithms and their convergence

This paper describes the application to three areas of physics of computer programs that carry out formal algebraic manipulation. The application areas discussed are celestial mechanics, general relativity and quantum electrodynamics. The paper describes typical problems from each of these disciplines which can be solved using algebraic manipulative systems and presents sample programs for the solution of these problems using several algebra systems. For each discipline a review of published work acknowledging the use of algebra programs is presented and the most advanced applications are discussed in detail. In particular the Lie transform, Petrov classification and Kahane's simplification procedure are reviewed from the standpoint of algebra programs. A number of simple examples are used to introduce the reader to the capabilities of an algebra program and a brief review of the technical problems of algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review. This review was completed in December 1971.

http://iopscience.iop.org/article/10.1088/0034-4885/35/1/305/pdf

Applications of algebraic manipulation programs in physics

The sheer magnitude of the work involved in the construction of perturbation theories (the “astronomical computations”) made it inevitable that astronomers would very early become interested in the possibility of constructing them with the aid of computers. Since then general systems for symbolic manipulations have been developed and used widely. Nevertheless there still remain problems for which these general systems are not well adapted and many specialized systems of algebraic manipulation (mostly Poisson series processors) are in use. An attempt is made to review this field by sketching some of the ideas on which these Poisson series processors are built.

A Survey of Poisson Series Processors

TRIP is an interactive computer algebra system that is devoted to perturbation series computations, and specially adapted to celestial mechanics Its development started in 1988, as an upgrade of the special purpose FORTRAN routines elaborated by J Laskar for the demonstration of the chaotic behavior of the Solar System TRIP is now a mature and efficient tool for handling multivariate generalized power series, and contains also a numerical kernel A limited version of TRIP is freely available on the WEB at wwwimccefr/trip

https://www.issac-conference.org/2010/assets/SoftwareDemos/GastineauFinal.pdf

TRIP: a computer algebra system dedicated to celestial mechanics and perturbation series

A programming system is described for the manipulation of Poisson series on a computer. The general structure is described together with the most important individual subroutines. The system exists in two versions, one with 3 angular and 3 polynomial variables and the other with twice as many variables. The programming has been done first in FORTRAN IV, but the most crucial subroutines have been rewritten in machine language for more efficiency. The storage mechanism is such that obsolete series can be erased from the main memory in order to make room for new series to be created.

K. Garthwaite

Papers

A programming system for analytical series expansions on a computer