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

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, (1978). Google Scholar
[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, (2009). Google Scholar
[3]	J. E. Marsden, Lectures on Mechanics,, London Math. Soc. Lecture Note Ser., (1992). doi: 10.1017/CBO9780511624001. Google Scholar
[4]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A basic exposition of Classical Mechanical Systems,, Texts in Applied Mathematics, (1994). doi: 10.1007/978-1-4612-2682-6. Google Scholar
[5]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Systems and the N-body Problem,, Applied Mathematical Sciences, (2009). Google Scholar
[6]	K. Meyer, Periodic Solutions of the N-Body Problem,, Lecture Notes in Mathematics, (1719). doi: 10.1007/BFb0094677. Google Scholar
[7]	K. Meyer and D. Schmidt, From the restricted to the full three-body problem,, Transactions AMS, 352 (2000), 2283. doi: 10.1090/S0002-9947-00-02542-3. Google Scholar
[8]	G. W. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift,, J. Nonlinear Science, 5 (1995), 373. doi: 10.1007/BF01212907. Google Scholar

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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, (1978). Google Scholar
[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, (2009). Google Scholar
[3]	J. E. Marsden, Lectures on Mechanics,, London Math. Soc. Lecture Note Ser., (1992). doi: 10.1017/CBO9780511624001. Google Scholar
[4]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A basic exposition of Classical Mechanical Systems,, Texts in Applied Mathematics, (1994). doi: 10.1007/978-1-4612-2682-6. Google Scholar
[5]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Systems and the N-body Problem,, Applied Mathematical Sciences, (2009). Google Scholar
[6]	K. Meyer, Periodic Solutions of the N-Body Problem,, Lecture Notes in Mathematics, (1719). doi: 10.1007/BFb0094677. Google Scholar
[7]	K. Meyer and D. Schmidt, From the restricted to the full three-body problem,, Transactions AMS, 352 (2000), 2283. doi: 10.1090/S0002-9947-00-02542-3. Google Scholar
[8]	G. W. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift,, J. Nonlinear Science, 5 (1995), 373. doi: 10.1007/BF01212907. Google Scholar

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