Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Field equations

Any equation of the form

ttu[t, x] xxu[t, 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

((tu)2 - (xu)2)/2 - v[u]

and conserves the Hamiltonian (energy function)

Integrate[((tu)2 + (tu)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].

Image Source Notebooks:

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