## A Modern Approach to Classical MechanicsThe approach to classical mechanics adopted in this book includes recent developments in nonlinear dynamical systems, and the concepts necessary to formulate and understand chaotic behavior are presented. Besides the conventional topics (such as oscillators, the Kepler problem, spinning tops and the two centers problem) studied in the frame of Newtonian, Lagrangian, and Hamiltonian mechanics, nonintegrable systems (the Henon -- Heiles system, the Coulomb force and the homogeneous magnetic field, the restricted three-body problem) are also discussed. The question of the integrability (of planetary motion, for example) leads finally to the KAM-theorem. This book is the result of lectures on 'Classical Mechanics' as the first part of a basic course in Theoretical Physics. These lectures were given by the author to undergraduate students in their second year at the Johannes Kepler University Linz, Austria. The book is also addressed to lecturers in this field and to physicists who want to obtain a new perspective on classical mechanics. |

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two fixed cnetres!!

### Contents

Introduction | 1 |

The foundations of mechanics | 11 |

Onedimensional motion of a particle | 27 |

Encountering peculiar motion in two dimensions | 59 |

Motion in a central force field | 85 |

The gravitational interaction of two bodies | 119 |

Collisions of particles Scattering | 145 |

Changing the frame of reference | 167 |

The rigid body | 249 |

Small oscillations | 291 |

Hamiltons canonical formulation of mechanics | 317 |

HamiltonJacobi theory | 347 |

From integrable to nonintegrable systems | 381 |

In retrospect | 411 |

B Rotations and tensors | 421 |

437 | |

### Common terms and phrases

1/r potential action-angle variables angular momentum angular velocity axis basis vectors body-fixed canonical equations canonical transformation center of mass central force field chaotic behavior classical mechanics components conservation law conserved quantities consider const constant constraint coordinate system curve degrees of freedom depends derivative determined differential equations dynamical eigenvalues eigenvectors ellipse equation of motion equilibrium positions Euler-Lagrange equation example fixed frame of reference frequency function given in Eq Hamiltonian harmonic oscillator Hence homogeneous inertial frame infinitesimal initial values integral interaction invariant Lagrange's equations Lagrangian linear magnetic field matrix momenta Newton's equation obtain orbit parameters particle pendulum perpendicular phase space physical plane Poincare section point masses problem radius vector reference frame relation relative motion respect rigid body rotating frame scattering angle shown in Fig solution spherical Subsection surface symmetry three body problem tion torus trajectory turning point two-dimensional vanishes yields