An approximation theorem in classical mechanics

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September 2016, 8(3): 359-374. doi: 10.3934/jgm.2016011

An approximation theorem in classical mechanics

Cristina Stoica ^1,

1.	Department of Mathematics, Wilfrid Laurier University, 75 University Ave West, Waterloo, ON, N2L 3C5

Received July 2015 Revised April 2016 Published September 2016

Abstract
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A theorem by K. Meyer and D. Schmidt says that The reduced three-body problem in two or three dimensions with one small mass is approximately the product of the restricted problem and a harmonic oscillator [7]. This theorem was used to prove dynamical continuation results from the classical restricted circular three-body problem to the three-body problem with one small mass.
We state and prove a similar theorem applicable to a larger class of mechanical systems. We present applications to spatial $(N+1)$-body systems with one small mass and gravitationally coupled systems formed by a rigid body and a small point mass.

Keywords: Restricted problems in mechanics, Lie symmetries, continuation of dynamics, reduction..

Mathematics Subject Classification: Primary: 37J20, 70H33; Secondary: 53D2.

Citation: Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition. With the assistance of T. Ratiu and R. Cushman, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	D. D. Holm, T. Schmah and C. Stoica, Geometry, Symmetry and Mechanics: From Finite to Infinite-dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, 2009.
[3]	J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Note Ser., 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.
[4]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A basic exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.
[5]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Systems and the N-body Problem, Applied Mathematical Sciences, 90, $2^{nd}$ edition, Springer-Verlag, New York, 2009.
[6]	K. Meyer, Periodic Solutions of the N-Body Problem, Lecture Notes in Mathematics, 1719, Springer-Verlag, New York, 1999. doi: 10.1007/BFb0094677.
[7]	K. Meyer and D. Schmidt, From the restricted to the full three-body problem, Transactions AMS, 352 (2000), 2283-2299. doi: 10.1090/S0002-9947-00-02542-3.
[8]	G. W. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift, J. Nonlinear Science, 5 (1995), 373-418. doi: 10.1007/BF01212907.

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References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition. With the assistance of T. Ratiu and R. Cushman, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	D. D. Holm, T. Schmah and C. Stoica, Geometry, Symmetry and Mechanics: From Finite to Infinite-dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, 2009.
[3]	J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Note Ser., 174, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.
[4]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A basic exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.
[5]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Systems and the N-body Problem, Applied Mathematical Sciences, 90, $2^{nd}$ edition, Springer-Verlag, New York, 2009.
[6]	K. Meyer, Periodic Solutions of the N-Body Problem, Lecture Notes in Mathematics, 1719, Springer-Verlag, New York, 1999. doi: 10.1007/BFb0094677.
[7]	K. Meyer and D. Schmidt, From the restricted to the full three-body problem, Transactions AMS, 352 (2000), 2283-2299. doi: 10.1090/S0002-9947-00-02542-3.
[8]	G. W. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift, J. Nonlinear Science, 5 (1995), 373-418. doi: 10.1007/BF01212907.

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