Mechanics and Dynamical Systems

Physics in 18 chapters — from “Newton's second law as an equation of motion” to “The Legendre transformation, the Hamiltonian and Hamilton's equations”.

Chapters

  1. Newton's second law as an equation of motion
    • The equation of motion
    • Separable equation of motion
    • Linear drag and the approach to terminal velocity
    • Quadratic drag and why heavy bodies fall faster
    • Position-dependent force
    • Problems
  2. Circular motion and angular kinematics
    • Uniform circular motion and centripetal acceleration
    • Plane polar coordinates
    • Velocity and acceleration in polar coordinates
    • Angular velocity and angular acceleration
    • Non-uniform circular motion
    • Problems
  3. Centre of mass, many-particle systems and rockets
    • Centre of mass
    • Centre-of-mass theorem and total momentum
    • The centre-of-mass frame
    • Variable-mass systems and the rocket equation
    • Using the rocket equation
    • Problems
  4. Conservative forces, potential energy and energy diagrams
    • Work as a line integral
    • Conservative forces and potential energy
    • Force from the gradient of the potential
    • Conservation of mechanical energy
    • Energy diagrams
    • Small-oscillation frequency from curvature
    • Problems
  5. Collisions in the laboratory and centre-of-mass frames
    • Impulse and the collision approximation
    • One-dimensional collisions
    • Kinetic energy split about the centre of mass
    • Collision in the centre-of-mass frame
    • Two-dimensional elastic collisions
    • Problems
  6. Statics and the equilibrium of forces and torques
    • Cross product and torque
    • Conditions of static equilibrium
    • Centre of gravity
    • Friction-limited equilibrium and tipping
    • Problems
  7. Rotation about a fixed axis, angular momentum and rolling
    • Moment of inertia and rotational kinetic energy
    • Equation of motion for fixed-axis rotation
    • Angular momentum of a particle and a system
    • Angular velocity vector and rigid-body angular momentum
    • Conservation of angular momentum
    • Rolling without slipping
    • Problems
  8. Simple, damped and driven oscillators
    • The simple harmonic oscillator
    • Energy in simple harmonic motion and the pendulum
    • Superposition and the damped oscillator
    • The quality factor and the phase plane
    • Driven oscillator steady state and resonance
    • Transients and linear response
    • Problems
  9. Coupled oscillators, normal modes and normal coordinates
    • Coupled equations of motion
    • Normal modes and the determinant condition
    • Normal coordinates
    • General motion as a superposition of modes
    • Unequal masses and the double pendulum
    • Problems
  10. Universal gravitation, the shell theorem and circular orbits
    • Newton's law of gravitation
    • Gravitational potential energy and potential
    • The shell theorem
    • Gravity inside a uniform Earth
    • Inertial and gravitational mass
    • Circular orbits and Kepler's third law
    • Geostationary orbit, escape velocity and orbital energy
    • Problems
  11. The two-body problem, reduced mass, the effective potential and the virial theorem
    • Two-body reduction and reduced mass
    • Central forces and angular momentum conservation
    • The radial equation and the effective potential
    • Classification of orbits by energy
    • Stability of circular orbits and radial oscillations
    • The virial theorem and cluster masses
    • Problems
  12. Kepler's laws, inverse-square orbits and orbital energetics
    • The orbit equation and its solution
    • Conic-section orbits and the ellipse
    • Kepler's second law and orbital periods
    • Orbital energy and the semi-major axis
    • Hohmann transfer, gravitational flyby and binaries
    • Problems
  13. Linearly accelerating frames and fictitious forces
    • Inertial frames and the Galilean transformation
    • Fictitious force in an accelerating frame
    • Effective gravity in accelerating vehicles
    • Einstein's lift and the equivalence principle
    • Problems
  14. Rotating frames: centrifugal and Coriolis forces, the Foucault pendulum and tides
    • Time derivative in a rotating frame
    • Equation of motion in a rotating frame
    • Effective gravity on the rotating Earth
    • Coriolis force, cyclones and falling bodies
    • The Foucault pendulum
    • Tidal forces and tides
    • Problems
  15. The calculus of variations and Hamilton's principle
    • Total time derivative of f(q,q˙​,t)
    • Functionals and the length of a path
    • The Euler-Lagrange equation
    • Shortest paths, the lifeguard and the brachistochrone
    • Hamilton's principle and the Lagrangian
    • Equivalence of Hamilton's principle with Newton's second law
    • Problems
  16. Generalised coordinates, constraints and Lagrange's equations
    • Degrees of freedom, constraints and generalised coordinates
    • Why constraint forces disappear
    • Lagrange's equations and the recipe
    • Worked Lagrangians for the Atwood machine and wedge
    • The two-body problem in Lagrangian form
    • The double pendulum in Lagrangian form
    • The bead on a rotating hoop
    • Problems
  17. Symmetry, cyclic coordinates, conserved momenta and Noether's theorem
    • Cyclic coordinates and conserved generalised momenta
    • Translation symmetry and linear momentum
    • Rotation symmetry and angular momentum
    • Time-translation symmetry and the Jacobi integral
    • Noether's theorem
    • When the conserved quantity is not the energy
    • Problems
  18. The Legendre transformation, the Hamiltonian and Hamilton's equations
    • The Legendre transform from velocity to momentum
    • The Hamiltonian and Hamilton's equations
    • When the Hamiltonian is the energy
    • Phase space
    • The oscillator in phase space
    • The two-body Hamiltonian in polar coordinates
    • What the canonical structure leads to
    • Phase portrait of the pendulum
    • Problems