Perturbation method in the theory of nonlinear oscillations
01 Mar 1970-Celestial Mechanics and Dynamical Astronomy (Kluwer Academic Publishers)-Vol. 3, Iss: 1, pp 90-106
TL;DR: Asymptotic recurrence formulas for treating nonlinear oscillation problems are presented in this paper, which are based on a Lie transform similar to that described by Deprit for Hamiltonian systems.
Abstract: Asymptotic recurrence formulas for treating nonlinear oscillation problems are presented These formulas are based on a Lie transform similar to that described by Deprit for Hamiltonian systems It is shown that the basic formulas have essentially the same forms as those obtained by Deprit and by the present author in the Hamiltonian case
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TL;DR: In this article, the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt and perturbation analysis is applied to this entire form, in all of its components.
Abstract: The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt. The perturbation analysis is applied to this entire form, in all of its components.
203 citations
TL;DR: In this article, the authors show how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm, and complete Kamel' contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.
Abstract: Kamel has recently extended to non-Hamiltonian equations a perturbation theory using Lie transforms. We show here how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm. Then we complete Kamel's contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.
117 citations
TL;DR: In this article, the quantum normal form of the Birkhoff-Gustavson normal form has been shown to be equivalent to the Rayleigh-Schrodinger perturbation.
Abstract: A quantum analog, called the quantum normal form, of the classical Birkhoff–Gustavson normal form is presented. The algebraic relationship between the quantum and Birkhoff–Gustavson normal forms has been established by developing the latter using Lie transforms. It is shown that the Birkhoff–Gustavson normal form can be obtained from the quantum normal form. Using an anharmonic oscillator and a Henon–Heiles system as test cases, the equivalence between the quantum normal form and the Rayleigh–Schrodinger perturbation method is shown. This equivalence provides an algebraic connection between the Birkhoff–Gustavson normal form and the Rayleigh–Schrodinger perturbation approach. The question of Weyl and torus quantizations of the Birkhoff–Gustavson normal form is discussed in the light of the quantum normal form.
70 citations
TL;DR: The application of algebraic manipulation in physics is discussed in this paper, where the application areas discussed are celestial mechanics, general relativity and quantum electrodynamics, and typical problems from each of these disciplines can be solved using algebraic manipulators.
Abstract: 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.
61 citations
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Book•
01 Jan 1961
TL;DR: In this article, a wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering.
Abstract: : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.
2,259 citations
TL;DR: In this paper, the concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter, and the formalism generates nonconservative as well as conservative transformations.
Abstract: 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.
880 citations
587 citations
TL;DR: In this article, the authors show how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm, and complete Kamel' contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.
Abstract: Kamel has recently extended to non-Hamiltonian equations a perturbation theory using Lie transforms. We show here how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm. Then we complete Kamel's contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.
117 citations
TL;DR: In this article, the theory of perturbation based on Lie transforms is considered and a simplified general recursion formulae are generated to speed up the implementation of such perturbations in the computerized symbolic manipulation.
Abstract: The theory of perturbation based on Lie transforms is considered. Deprit's equation is reduced to a form which enables us to generate simplified general recursion formulae. These expansions are then modified to speed up the implementation of such perturbation theory in the computerized symbolic manipulation.
104 citations