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Classical Mechanics - University of California, Los lecture notes...6.6 Relation to a problem of geodesics on SO(N) ... 9.5 Solution by Lax pair ... with statistical mechanics, classical

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Text of Classical Mechanics - University of California, Los lecture notes...6.6 Relation to a problem of...

  • Classical Mechanics

    Eric DHokerDepartment of Physics and Astronomy,

    University of California, Los Angeles, CA 90095, USA

    2 September 2012

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  • Contents

    1 Review of Newtonian Mechanics 41.1 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Newtons laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Comments on Newtons laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Dissipative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Velocity dependent conservative forces . . . . . . . . . . . . . . . . . . . . . 101.8 Charged particle in the presence of electro-magnetic fields . . . . . . . . . . 121.9 Physical relevance of conservative forces . . . . . . . . . . . . . . . . . . . . 13

    2 Lagrangian Formulation of Mechanics 142.1 The Euler-Lagrange equations in general coordinates . . . . . . . . . . . . . 142.2 The action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Euler-Lagrange equations from the action principle . . . . . . . . . . . . . . 202.5 Equivalent Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Systems with constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7 Holonomic versus non-holonomic constrains . . . . . . . . . . . . . . . . . . 23

    2.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7.2 Reducibility of certain velocity dependent constraints . . . . . . . . . 25

    2.8 Lagrangian formulation for holonomic constraints . . . . . . . . . . . . . . . 262.9 Lagrangian formulation for some non-holonomic constraints . . . . . . . . . . 282.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.11 Symmetry transformations and conservation laws . . . . . . . . . . . . . . . 302.12 General symmetry transformations . . . . . . . . . . . . . . . . . . . . . . . 322.13 Noethers Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.14 Examples of symmetries and conserved charges . . . . . . . . . . . . . . . . 35

    2.14.1 Translation in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.14.2 Translation in a canonical variable . . . . . . . . . . . . . . . . . . . 35

    3 Quadratic Systems: Small Oscillations 373.1 Equilibrium points and Mechanical Stability . . . . . . . . . . . . . . . . . . 373.2 Small oscillations near a general solution . . . . . . . . . . . . . . . . . . . . 403.3 Lagrange Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Stability near the non-colinear Lagrange points . . . . . . . . . . . . . . . . 43

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  • 4 Hamiltonian Formulation of Mechanics 454.1 Canonical position and momentum variables . . . . . . . . . . . . . . . . . . 454.2 Derivation of the Hamiltons equations . . . . . . . . . . . . . . . . . . . . . 464.3 Some Examples of Hamiltonian formulation . . . . . . . . . . . . . . . . . . 474.4 Variational Formulation of Hamiltons equations . . . . . . . . . . . . . . . . 484.5 Poisson Brackets and symplectic structure . . . . . . . . . . . . . . . . . . . 494.6 Time evolution in terms of Poisson brackets . . . . . . . . . . . . . . . . . . 504.7 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 Symmetries and Noethers Theorem . . . . . . . . . . . . . . . . . . . . . . . 534.9 Poissons Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.10 Noether charge reproduces the symmetry transformation . . . . . . . . . . . 54

    5 Lie groups and Lie algebras 565.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Matrix multiplication groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Orthonormal frames and parametrization of SO(N) . . . . . . . . . . . . . . 595.4 Three-dimensional rotations and Euler angles . . . . . . . . . . . . . . . . . 615.5 Definition of a Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.6 Definition of a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.7 Relating Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . . 635.8 Symmetries of the degenerate harmonic oscillator . . . . . . . . . . . . . . . 645.9 The relation between the Lie groups SU(2) and SO(3) . . . . . . . . . . . . 67

    6 Motion of Rigid Bodies 686.1 Inertial and body-fixed frames . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Kinetic energy of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Angular momentum of a rigid body . . . . . . . . . . . . . . . . . . . . . . . 706.4 Changing frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.5 Euler-Lagrange equations for a freely rotating rigid body . . . . . . . . . . . 726.6 Relation to a problem of geodesics on SO(N) . . . . . . . . . . . . . . . . . 736.7 Solution for the maximally symmetric rigid body . . . . . . . . . . . . . . . 746.8 The three-dimensional rigid body in terms of Euler angles . . . . . . . . . . 746.9 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.10 Poinsots solution to Eulers equations . . . . . . . . . . . . . . . . . . . . . 76

    7 Special Relativity 787.1 Basic Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Lorentz vector and tensor notation . . . . . . . . . . . . . . . . . . . . . . . 807.3 General Lorentz vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . 82

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  • 7.3.1 Contravariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.3.2 Covariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.3.3 Contraction and trace . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    7.4 Relativistic invariance of the wave equation . . . . . . . . . . . . . . . . . . . 847.5 Relativistic invariance of Maxwell equations . . . . . . . . . . . . . . . . . . 85

    7.5.1 The gauge field, the electric current, and the field strength . . . . . . 857.5.2 Maxwells equations in Lorentz covariant form . . . . . . . . . . . . . 86

    7.6 Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.7 Relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.8 Lagrangian for a massive relativistic particle . . . . . . . . . . . . . . . . . . 917.9 Particle collider versus fixed target experiments . . . . . . . . . . . . . . . . 927.10 A physical application of time dilation . . . . . . . . . . . . . . . . . . . . . 93

    8 Manifolds 948.1 The Poincare Recurrence Theorem . . . . . . . . . . . . . . . . . . . . . . . 948.2 Definition of a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.4 Maps between manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.5 Vector fields and tangent space . . . . . . . . . . . . . . . . . . . . . . . . . 978.6 Poisson brackets and Hamiltonian flows . . . . . . . . . . . . . . . . . . . . . 998.7 Stokess theorem and grad-curl-div formulas . . . . . . . . . . . . . . . . . . 1008.8 Differential forms: informal definition . . . . . . . . . . . . . . . . . . . . . . 1028.9 Structure relations of the exterior differential algebra . . . . . . . . . . . . . 1038.10 Integration and Stokess Theorem on forms . . . . . . . . . . . . . . . . . . . 1058.11 Frobenius Theorem for Pfaffian systems . . . . . . . . . . . . . . . . . . . . . 105

    9 Completely integrable systems 1079.1 Criteria for integrability, and action-angle variables . . . . . . . . . . . . . . 1079.2 Standard examples of completely integrable systems . . . . . . . . . . . . . . 1079.3 More sophisticated integrable systems . . . . . . . . . . . . . . . . . . . . . . 1089.4 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.5 Solution by Lax pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.6 The Korteweg de Vries (or KdV) equation . . . . . . . . . . . . . . . . . . . 1129.7 The KdV soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.8 Integrability of KdV by Lax pair . . . . . . . . . . . . . . . . . . . . . . . . 114

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  • Bibliography

    Course textbook

    Classical Dynamics (a contemporary perspective),J.V. Jose and E.J. Saletan, Cambridge University Press, 6-th printing (2006).

    Classics

    Mechanics, Course of Theoretical Physics, Vol 1, L.D. Landau and E.M. Lifschitz,Butterworth Heinemann, Third Edition (1998).

    Classical Mechanics, H. Goldstein, Addison Wesley, (1980);

    Further references

    The variational Principles of Mechanics, Cornelius Lanczos, Dover, New York (1970). Analytical Mechanics, A. Fasano and S. Marmi, Oxford Graduate Texts.

    More Mathematically oriented treatments of Mechanics

    Mathematical Methods of Classical Mechanics, V.I. Arnold, Springer Verlag (1980).

    Foundations of Mechanics, R. Abraham and J.E. Marsden, Addison-Wesley (1987).

    Higher-dimensional Chaotic and Attractor Systems, V.G. Ivancevic and T.T. Ivancevic,Springer Verlag.

    Physics for Mathematicians, Mechanics I; by Michael Spivak (2011).

    Mathematics useful for physics

    Manifolds, Tensor Analysis, and Applications, R. Abraham, J.E. Marsden, T. Ratiu,Springer-Verlag (1988).

    Geometry, Topology and Physics, M. Nakahara, Ins