Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Field equations

Any equation of the form

∂[u[t, x], t, t] ==∂[u[t, x], x, x] + f[u[t, x]]

can be thought of as a classical field equation for a scalar field. Defining

v[u] = -Integrate[f[u], u]

the field then has Lagrangian density

(∂[u, t]2 - ∂[u, x]2)/2 - v[u]

and conserves the Hamiltonian (energy function)

Integrate[(∂[u, t]2 + ∂[u, x]2)/2 + v[u], {x, -∞, ∞}]

With the choice for f[u] made here (with a >= 0), v[u] is bounded from below, and as a result it follows that no singularities ever occur in u[t, x].

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From Stephen Wolfram: A New Kind of Science [citation]