Mechanics

Force, momentum, energy, rotation, the continuum limit, and the variational view.

Mechanics is the science of motion under force. Newton’s three laws fix how a body responds to force; integrating them over time gives momentum and impulse, over a path gives work and energy, and across a system gives the rotational laws and the conditions for equilibrium. Taking the laws to a continuum produces the field equations of fluids and solids, and reformulating them as a variational principle gives the analytical mechanics on which the rest of theoretical physics is built.

• 1.1 Force and Newton’s three laws — inertial frames, $\mathbf F = d\mathbf p/dt$, the third law, systems and the centre of mass, free-body diagrams.
• 1.2 Momentum, impulse, and collisions — the impulse–momentum theorem, momentum conservation, elastic and inelastic collisions, the restitution coefficient.
• 1.3 Work, energy, and conservation — the work–energy theorem, conservative forces and potential energy, conservation of mechanical energy, power.
• 1.4 Rotation: torque, angular momentum, statics — $\boldsymbol\tau = d\mathbf L/dt$, the moment of inertia, equilibrium of force and torque, the lever.
• 1.5 From particles to continua — the mass–spring chain, the continuum limit and the wave equation, and the master equation $\rho\,D\mathbf u/Dt = \nabla\cdot\boldsymbol\sigma$.
• 1.6 The variational view: Lagrangian mechanics — the action, the Euler–Lagrange equation, generalised coordinates and constraints, symmetry and conservation.