Notes

Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations


Field equations

Any equation of the form

D[u[t, x], t, t] ==D[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

(D[u, t]^2 - D[u, x]^2)/2 - v[u]

and conserves the Hamiltonian (energy function)

Integrate[(D[u, t]^2 + D[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].


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