## An Introduction to Lagrangian MechanicsAn Introduction to Lagrangian Mechanics begins with a proper historical perspective on the Lagrangian method by presenting Fermat's Principle of Least Time (as an introduction to the Calculus of Variations) as well as the principles of Maupertuis, Jacobi, and d'Alembert that preceded Hamilton's formulation of the Principle of Least Action, from which the Euler–Lagrange equations of motion are derived. Other additional topics not traditionally presented in undergraduate textbooks include the treatment of constraint forces in Lagrangian Mechanics; Routh's procedure for Lagrangian systems with symmetries; the art of numerical analysis for physical systems; variational formulations for several continuous Lagrangian systems; an introduction to elliptic functions with applications in Classical Mechanics; and Noncanonical Hamiltonian Mechanics and perturbation theory.This textbook is suitable for undergraduate students who have acquired the mathematical skills needed to complete a course in Modern Physics. |

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### Contents

The Calculus of Variations | 1 |

Lagrangian Mechanics | 35 |

Hamiltonian Mechanics | 73 |

Motion in a CentralForce Field | 95 |

Collisions and Scattering Theory | 119 |

Motion in a NonInertial Frame | 141 |

Rigid Body Motion | 159 |

NormalMode Analysis | 187 |

Continuous Lagrangian Systems | 203 |

Appendix A Basic Mathematical Methods | 217 |

Appendix B Elliptic Functions and Integrals | 229 |

Noncanonical Hamiltonian Mechanics | 245 |

255 | |

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### Common terms and phrases

acceleration angle angular momentum angular velocity axis Calculus of Variations canonical central potential central-force CM frame collision components conservation law consider constant constraint coordinates Coriolis cross section deﬁned deﬁnition denotes derivative dynamics equations of motion equilibrium point Euler equations Euler-Lagrange equation expressed in terms ﬁeld ﬁnd ﬁrst ﬁxed Foucault pendulum Frenet-Serret Frenet-Serret formulas frequency gravitational Hamilton’s Hamiltonian Hamiltonian mechanics Hence inertia tensor initial conditions integral Kepler problem kinetic energy LAB frame Lagrange Lagrangian density Lagrangian Mechanics Lastly Legendre transformation light ray linear masses m1 massless matrix Note obtain oscillation parameter particle of mass path perturbation position potential energy potential U(r precession Principle of Least radial reﬂected rotating frame satisﬁes scattering separatrix Show sin2 solution symmetric top transformation turning points unit vector vanishes variational principle Weierstrass elliptic function yields