Abstract
In this investigation, a procedure is described for extending the application of canonical perturbation theories, which have been applied previously to the study of conservative systems only, to the study of non-conservative dynamical systems. The extension is obtained by imbedding then-dimensional non-conservative motion in a 2n-dimensional space can always be specified in canonical form, and, consequently, the motion can be studied by direct application of any canonical perturbation method. The disadvantage of determining a solution to the 2n-dimensional problem instead of the originaln-dimensional problem is minimized if the canonical transformation theory is used to develop the perturbation solution. As examples to illustrate the application of the method, Duffing's equation, the equation for a linear oscillator with cubic damping and the van der Pol equation are solved using the Lie-Hori perturbation algorithm.
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References
Bogoliubov, N. and Mitropolsky, Y.: 1961,Asymptotic Method in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.
Brouwer, D.: 1966, ‘Solution of the Problem of Artificial Satellite Theory Without Drag,’Astron. J. 64, 378.
Brouwer, D. and Hori G.-I.: 1961, ‘Theoretical Evaluation of Atmospheric Drag Effects in the Motion of an Artificial Satellite’,Astron. J. 66, 193.
Campbell, J. A. and Jefferys, W. H.: 1970, ‘Equivalence of the Perturbation Theories of Hori and Deprit’,Celest. Mech. 2, 467.
Deprit, A.: 1969, ‘Canonical Transformations Depending on a Small Parameter’,Celest. Mech. 1, 12.
Deprit, A. and Rom, A.: 1967, ‘Asymptotic Representation of the Cycle of van der Pol's Equation for Small Damping Coefficients,’Z. angew. Math. Physik,18, 736.
Hildebrand, C. H., Jr.: 1969, ‘A Discussion of Two General Perturbation Methods and Their Application to Artificial Satellite Theory’, TR-1004, Applied Mechanics Research Laboratory, The University of Texas at Austin, Austin, Texas.
Hori, G-I.: 1966, ‘Theory of General Perturbations with Unspecified Canonical Variables’,Astron. Soc. Japan,18, 4.
Hori, G-I.: 1971, ‘Theory of General Perturbation for Non-Canonical Systems’,Publ. Astron. Soc. Japan,23, 567.
Jefferys, H.: 1970, ‘TRIGMAN-A System for Algebraic Manipulations of Poisson Series’, Applied Mechanics Research Laboratory, Report No. AMRL 1032, The University of Texas at Austin, Austin, Texas (August 1970).
Kamel, A.: 1970, ‘Perturbation Method in the Theory of Nonlinear Oscillations’,Celest. Mech. 3, 90.
Kamel, A. A.: 1971, ‘Lie Transforms and the Hamiltonization of Non-Hamiltonian Systems’,Celest. Mech. 4, 397.
Lane, M. H. and Crawford, K. H.: 1969, ‘An Improved Analytical Drag Theory for the Artificial Satellite Problem’, AIAA/AAS Astrodynamics Conference, AIAA Paper 69-925.
Mersman, W. A.: 1970a, ‘A New Algorithm for the Lie Transformation’,Celest. Mech. 3, 81.
Mersman, W. A.: 1970b, ‘Explicit Recessive Algorithms for the Construction of Equivalent Canonical Transformations’,Celest. Mech. 3, 384.
Powers, W. F. and McDonell, J. P.: 1970, ‘Switching Conditions and a Synthesis Technique for the Singular Saturn Guidance Problem’, AIAA Guidance, Control and Flight Mechanics Conference, AIAA Paper No. 70-965, Santa Barbara, California, August, 1970.
Powers, W. F. and Tapley, B. D.: 1969, ‘Canonical Applications to Optimal Trajectory Analysis’,AIAA J. 7, 394.
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This research was supported by the Office of Naval Research under Contract N00014-67-a-0126-0013.
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Choi, J.S., Tapley, B.D. An extended canonical perturbation method. Celestial Mechanics 7, 77–90 (1973). https://doi.org/10.1007/BF01243509
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DOI: https://doi.org/10.1007/BF01243509