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

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.
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

show all references

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

[1]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

[2]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014

[3]

Henrique Bursztyn, Alejandro Cabrera. Symmetries and reduction of multiplicative 2-forms. Journal of Geometric Mechanics, 2012, 4 (2) : 111-127. doi: 10.3934/jgm.2012.4.111

[4]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[5]

Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312

[6]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

[7]

Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial & Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023

[8]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[9]

Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99

[10]

Alain Miranville, Mazen Saad, Raafat Talhouk. Preface: Workshop in fluid mechanics and population dynamics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : i-i. doi: 10.3934/dcdss.2014.7.2i

[11]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291

[12]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541

[13]

Juan Belmonte-Beitia, Víctor M. Pérez-García, Vadym Vekslerchik, Pedro J. Torres. Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 221-233. doi: 10.3934/dcdsb.2008.9.221

[14]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347

[15]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[16]

Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049

[17]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[18]

Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143

[19]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[20]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]