1.5 From particles to continua

Newton’s laws were stated for point masses, but most of physics governs continua — fluids, elastic solids, membranes, fields. The continuum form is the same mechanics at a different scale: Newton’s second law applied to a differential element, together with a constitutive law for how that element responds to deformation. The bridge between the particle and the field is worth building once explicitly, on the simplest system that shows it.

A chain of masses

Take $N$ identical masses $m$ at equilibrium positions $x_j = j a$ , each joined to its neighbours by springs of stiffness $\kappa$ . Let $u_j$ be the displacement of mass $j$ . The spring on its right is stretched by $u_{j+1} - u_j$ , the one on its left by $u_j - u_{j-1}$ , and Newton’s second law for mass $j$ reads

m\,\ddot u_j \;=\; \kappa\,(u_{j+1} - 2 u_j + u_{j-1}).

The right-hand side is a discrete second difference. Expanding the neighbours, $u_{j\pm1} = u(x_j\pm a) \approx u \pm a\,u_x + \tfrac12 a^2 u_{xx}$ , the second difference becomes $a^2\,u_{xx} + O(a^4)$ . Taking $a\to 0$ with the wave speed $c^2 = \kappa a^2/m$ held fixed collapses the chain of coupled ODEs into a single field equation, the wave equation:

\frac{\partial^2 u}{\partial t^2} \;=\; c^2\,\frac{\partial^2 u}{\partial x^2}, \qquad c = a\sqrt{\frac{\kappa}{m}}.

The discrete system of particles has become a continuous field obeying a partial differential equation. This is the prototype of every continuum theory: a microscopic lattice, a limit, and a field equation.

Number of masses N = 32

A pulse on a chain of masses on springs propagates as a wave. For small N the chain has dispersion — short-wavelength components travel slower than long ones — and the pulse spreads and ripples. As N grows, the chain's spectrum fills out smoothly and approaches the continuum wave equation u_tt = c²u_xx: the dashed reference. This is how Newton's particle-mechanics becomes continuum field theory.

A pulse on a chain of $N$ masses propagates as a wave. For small $N$ the chain is dispersive — short wavelengths lag — and the pulse spreads and ripples; as $N$ grows the chain’s behaviour approaches the smooth continuum solution (the dashed reference). The continuum wave equation is the long-wavelength limit of the discrete chain beneath it.

The master equation of continuum mechanics

The same logic runs in three dimensions and for any internal force law. A material element of volume $dV$ has mass $\rho\,dV$ ; the net force on it from its neighbours is the divergence of the stress tensor $\boldsymbol\sigma$ — the surface forces on its faces — times $dV$ . Newton’s second law per unit volume is then

\rho\,\frac{D\mathbf u}{Dt} \;=\; \nabla\cdot\boldsymbol\sigma,

with $D/Dt$ the rate of change following the material. This is the master equation of continuum mechanics. What distinguishes one continuum from another is only the constitutive law that gives $\boldsymbol\sigma$ : a pressure for an inviscid fluid, pressure plus a viscous stress for a real fluid, an elastic stress proportional to strain for a solid. The fluid mechanics and elasticity chapters are this one equation closed with two different constitutive laws.