CLASSICAL MECHANICS
Dimitar Trifonov (INRNE, "Avangard", Sofia 2002)
CONTENTS (extended)
Paragraphs and subparagraphs noted by star * may be omitted at first reading.
Contents (in Bulgarian)........................................................................... 3
Preface ............................................................................................ 5
Abbreviations .................................................................................. 6
Chapter 1. BASIC MECHANICAL AND GEOMETRIC NOTIONS
1. Basic notions of classical mechanics....................... ...................... 7
1.1. Basic notions. 1.2. Space and time. 1.3. Mechanical quantities.
1.4. State, mechanical laws and principles.
2. Vector and affine spaces ............................................................... 13
2.1. Vector space. Matrices as operators. Vectors and tensors. 2.2 Scalar
product and norm. Linear functions on vector space. External forms.* 2.3. Affine
space. Distance. Parallel transport*. Examples of affine spaces. Manifold.*
Differential forms.* Norm and distance.*
3. Cartesian, spherical and cylindrical coordinates............................ 29
3.1. Curvilinear coordinates in E_n. 3.2. Spherical and cylindrical
coordinates in E_3 .
4. Generalized coordinates, degrees of freedom, constraints ............ 34
4.1. Degrees of freedom and generalized coordinates. 4.2. Constraints.
Summary................................................................................. 37
Problems..................................................................................... 38
Chapter 2. LAGRANGIAN FORMULATION OF MECHANICS
5. Least action principle................................................................. 39
5.1.Formulation. 5.2. Lagrange equations. 5. 3. Freedom in the
choice of Lagrange function.
6. Lagrange equations for particle system. Reduced action ............... 45
6.1. Free particle. 6.2. Particle in external field. 6.3. N particles.
6.4. Generalized momenta and forces. 6.5.Generalized potential forces
and friction forces. 6.6. Reduced action. Examples. Historical notes. *
Summary.....................................................................................57
Chapter 3. INVARIANCE AND CONSERVATION LAWS
7. Invariance and covariance of equations and quantities .................. 59
7.1. Coordinate transformations and Lagrange equations. 7.2. Scalar
and invariant quantities. Examples of invariant and non-invariant quantities.
7.3. Covariance of Lagrange equations. Velocities and momenta belong to
different spaces.* 7.4. Absolute, relative and transferred generalized velocity.
8. Mechanical similarity. Virial theorem.......................................... 71
8.1. Scale transformations. 8.2. Virial theorem.
9. Conserved quantities in mechanics............................................. 74
9.1. Definition and number. 9.2.Conservation of energy. Generalized
energy and total energy. Difference in names. 9.3. Conservation of the momentum.
9.4. Conservation of the angular momentum. 9.5. Noether theorem. Cyclic
coordinates.
10. Transformation of E, P, and M. Relativity principle ................... 84
10.1. Translation. 10.2. Rotation.Group. Group of rotations.* 10.3. Motion with
constant velocity. 10.4. Relativity principle.
Summary.................................................................................... 91
Problems.................................................................................... 93
Chapter 4. ONE- AND TWO-DIMENSIONAL SYSTEMS
11. One-dimensional motion......................................................... 95
11.1. Free particle. 11.2. Harmonic oscillator. 11.3. Nonstationary
oscillator. 11.4. Pendulum. Non-cartesian coordinates.
12. Two-dimensional motion........................................................ 103
12.1. Two-dimensional oscillator.12.2. Two-dimensional pendulum.
Motion on a sphere.
13. Motion in a central field.......................................................... 108
13.1. Central field. 13.2. Consequences from the conservation of E
and M. 13.3. Closed trajectories and fall at the center. Symmetry of
the trajectories.
14. Kepler problem................................................................... 112
14.1. Coulomb field. 14.2. Conic cross-sections. Cartesian coordinates.
14.3. Kepler laws. Runge--Lentz vector.
Summary.................................................................................. 119
Problems.................................................................................. 120
Chapter 5. SYSTEMS OF PARTICLES
15. Two particle system............................................................. 123
15.1. Equations of motion in L-system. 15.2. Transition to C-system.
15.3. Example.
16. Elastic collisions of particles................................................... 126
16.1. Definition and conserved quantities. 16.2. Description in C-system.
16.3. Transition to L-system.
17. Particle scattering.................................................................. 130
17.1. Static center. Scattering cross-section. 17.2. Coulomb center.
Routherford formula. 17.3. Mobile targets. 17.4 Distributions.
Summary....................................................................................136
Chapter 6. OSCILLATIONS OF MECHANICAL SYSTEMS
18. Small oscillations ................................................................... 137
18.1. s degrees of freedom. Lagrange function and equations. 18.2. Eigen
frequencies. Normal coordinates (modes). 18.3. Connected pendulums.
19. Damped and forced oscillations .............................................. 144
19.1. Damped and forced oscillator. 19.2. Damped oscillator. Dissipative
function. Lagrange function for the damped oscillator.* 19.3. Forced oscillator.
Resonance.
20. Nonlinear oscillations............................................................ 150
20.1. Anharmonic oscillations. 20.2. Examples of anharmonic oscillations.
Summary................................................................................ 154
Problems............................................................................... 156
Chapter 7. RIGID BODY MOTION
21. Kinetic energy and tensor of inertia...................................... 157
21.1. Rigid body. Angular velocity. 21.2. Kinetic energy. Tensor of inertia.
21.3. Principal inertial moments and axes. 21.4. Transformation of inertia
tensor under rotation and translation.
22. Equations of motion of the rigid body................................... 167
22.1. Angular momentum of rigid body. 22.2. Free axes of rotation.
22.3. Lagrange function and equations ofmotion. 22.4. Transformation
of M and K under translation.
23. Euler angles. Top precession............................................. 174
23.1. Euler angles. 23.2. Top precession. 23.3. Symmetric top in the
field of Earth attraction.
24. Motion in a non-inertial frame............................................ 180
A. Particle motion. Inertial and non-inertial frames. Transformation of coordinates
and velocities. Lagrange function and equation of motion in K'. Vector form. Inertial
forces. Energy and angular momentum transformation.
B. Rigid body motion.
Summary....................................................................................... 192
Problems........................................................................................ 194
Chapter 8. HAMILTON APPROACH IN MECHANICS
25. Canonical equations................................................................... 195
25.1. Hamilton function. Example: N particles system. Legendre transform. *
25.2. Canonical equations. Historical notes. *
26. Time-development and Poisson brackets.................................... 204
26.1. Time-derivative of mechanical quantities. 26.2. Poisson brackets.
Examples of Poisson brackets. Algebra, generated by M_i .* 26.3. Poisson theorem.
27. Hamilton--Jacobi equation........................................................ 208
27.1. Action as a function of coordinates and time. 27.2. Hamilton--Jacobi
equation. Necessary and sufficient condition.
28. Canonical transformations ......................................................... 214
28.1. Canonical transformations. 28.2. Canonicity criteria. Generating functions
and Poisson brackets. Reconstruction of canonical transformation. Four examples.
28.3. Motion as a canonical transformation. 28.4. Coordinate and time scale
transformations as canonical ones.
29. Inverse problem * ................................................................... 235
29.1. Direct and inverse problem. 29.2. Reduction to initial values.
29.3. Linearization of Hamiltonian. Illustrative examples in solving the inverse
problem (free particle, harmonic oscillator).*
30. Phase space .............................................................................. 243
30.1. Phase space. Symplectic structure of phase space.* 30.2. Phase trajectories.
Liouville theorem. 30.3. Universal invariants. Volume form. Phase flow.*
Summary........................................................................................ 251
Problems......................................................................................... 253
Chapter 9. RELATIVISTIC MECHANICS
31. Poincare--Einstein relativity principle........................................... 255
31.1. Galilei and Lorentz transformations. 31.2. Lorentz transformations.
Lorentz group.* 31.3. Poincare--Einstein relativity principle.
31.4. Special
Lorentz transformations. General Lorentz transformations.* 31.5. Propertime
and length. Velocity transformations.
32. Space of events......................................................................... 267
32.1. Events. 32.2. Minkowski space. 32.3. Relativistic interval.
Light cone. 32.4. Four-dimensional vectors and tensors.
33. Relativistic particle mechanics.................................................... 272
33.1. Least action principle and equation of motion. 33.2. Energy and
momentum. Hamilton function. 33.3. Angular momentum. Minkowski
equation.
34. Tachyons * ............................................................................. 278
34.1. Causality principle. 34.2. Lagrange function, energy and momentum
of tachyon. 34.3. Switching principle. 34.4. Bradion-tachyon symmetry.*
34.5. Historical notes.
Summary...................................................................................... 283
Problems..................................................................................... 285
Appendix: ELEMENTS OF VARIATIONAL CALCULUS
A1. Functionals............................................................................. 286
A2. Extremum of a functional........................................................ 288
Examples: Length functionals.
A3. Action functionals................................................................... 294
A4. Dirac delta-function............................................................... 295
References................................................................................... 299
Notations ................................................................................... 301
Alphabetical index ........................................................................ 303
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