Lectures on Classical Mechanics

The following sections are included:

• Dedication
• Preface
• Contents
• Glossary

We begin with Kinematics, that is: How do we describe the physical objects under study? First, there is an idealization that can be removed later, in the simplification that we treat material bodies as if they were point-like objects. This should be understood as focusing on the relations between material bodies that are sufficiently far apart that their individual sizes and shapes do not matter: When dealing with the earth in orbit about the sun, the size of the earth is so tiny compared with its distance from the sun that we get a very good understanding of the physics by describing the earth as a point-like object with position and velocity — one position and one velocity, not many positions and velocities for all the parts that constitute the earth. Other phenomena, such as tides, for example, do require a much more detailed description of the earth, but even then we can think of her as being composed of a large number of essentially point-like parts. In short, it is well justified to begin with a discussion of the kinematics of point-like massive bodies, of mass points or point masses. The emphasis on mass, that is: on the material nature of the objects, is a reminder that we are not going to deal with immaterial things — the motion of shadows, for example, may be intriguing, but it is not what we are interested in.

The following sections are included:

• Newton's equation of motion
• Elementary examples
  • Force-free motion
  • Constant force
  • Frictional forces
  • Linear restoring force: Harmonic oscillations
  • Damped harmonic oscillations
  • Damped and driven harmonic oscillations
    • Periodic harmonic force: Resonance
    • Impulsive force: Heaviside's step function and Dirac's delta function
    • Arbitrary driving force: Green's function
    • Periodic impulsive force: Cyclically steady state

The following sections are included:

• One-dimensional motion
  • Kinetic and potential energy
  • Bounded motion between two turning points
    • Small-amplitude oscillations
    • Large-amplitude oscillations
    • Potential energy inferred from the energy-dependent period
  • Unbounded motion with a single turning point
  • Unbounded motion without a turning point
• Three-dimensional motion
  • Kinetic energy, potential energy
  • Conservative force fields
    • Necessity of vanishing curl
    • Sufficiency of vanishing curl — Stokes's theorem
  • Extremal points of the potential energy: Maxima, minima, saddle points
  • Potential energy in the vicinity of an extremal point
  • Example: Electrostatic potentials have no maxima or minima

The following sections are included:

• Reciprocal forces: Conservation of momentum
• Conservative pair forces: Conservation of energy
• Line-of-sight forces: Conservation of angular momentum
• Conservative line-of-sight forces
• Additional external forces
  • Transfer of momentum, energy, and angular momentum
  • Center-of-mass motion
  • Conservative external forces

The following sections are included:

• Center-of-mass motion and relative motion, reduced mass
• Kepler's ellipses and Newton's force law
• Motion in a central-force field
  • Bounded motion
  • Unbounded motion
  • Scattering

The following sections are included:

• Gravitational potential
• Monopole moment and quadrupole moment dyadic
• Newton's shell theorem
• Green's function of the Laplacian differential operator

The following sections are included:

• Johann Bernoulli's challenge: The brachistochrone
• Euler–Lagrange equations
• Solution of the brachistochrone problem
• Jakob Bernoulli's problem: The catenary
• Handling constraints: Lagrange multipliers

The following sections are included:

• Lagrange function
  • One coordinate
  • More coordinates
  • Change of description, cyclic coordinates
• Time and energy
• Examples
  • Two masses strung up
  • Two coupled harmonic oscillators

The following sections are included:

• Near an equilibrium: Lagrange function and equations of motion
• Characteristic frequencies and normal modes
• Examples
  • Pendulum
  • Double pendulum
  • Linear triatomic molecule

The following sections are included:

• Time as a coordinate
• Endpoint variations: Momentum and Hamilton function
• Five remarks
  • Natural variables
  • The minus sign
  • Legendre transformations
  • Many coordinates
  • Kinetic and canonical momentum
• Cyclic coordinates and constants of motion
  • Energy
  • Total momentum
  • Total angular momentum
• Hamilton's equations of motion
• Poisson bracket
• Conservation laws and symmetries. Noether's theorem
• One-dimensional motion; two-dimensional phase space
• Phase-space density. Liouville's theorem
• Velocity-dependent forces and Schwinger's action
• An excursion into the quantum realm

The following sections are included:

• Inertia dyadic. Steiner's theorem. Principal axes
• Euler's equation of motion
  • The general case
  • No torque acting
    • Two equal moments of inertia
    • Rotation about a principal axis
• Examples
  • Physical pendulum
  • Thin rod
  • Symmetric top

The following sections are included:

• Coriolis force, centrifugal force
• Examples
  • Foucault's pendulum
  • Deflection of a falling mass
  • Gyrocompass

The following sections are included:

• Exercises with Hints
  • Chapter 1
  • Chapter 2
  • Chapter 3
  • Chapter 4
  • Chapter 5
  • Chapter 6
  • Chapter 7
  • Chapter 8
  • Chapter 9
  • Chapter 10
  • Chapter 11
  • Chapter 12
  • Hints
• Appendix
  • A: On conic sections
    • A.1: Foci; vertices; cartesian coordinates; ray optics
    • A.2: Eccentricity; directrices; polar coordinates
    • A.3: Plane sections of a cone
  • B: On the exercise for the reader in Section 5.2
  • C: On the exercise for the reader in Section 11.2.2
• Index