Hamilton's principle/Mathematics/Several independent variables

Plain-language version: More than one variable

The last extension of the machinery replaces the interval by a domain and the single independent variable by several. The Euler-Lagrange equation becomes a partial differential equation, and the boundary term becomes a surface integral handled by the divergence theorem. This is the form in which every field theory in the applications track is written.

Setup

The boundary term in one dimension and in a field.

Let $\Omega \subset {ℝ}^m$ be a bounded domain with piecewise smooth boundary $\partial \Omega$, with points written $x=(x_1,\dots ,x_m)$. Let $u:\Omega \to ℝ$ be twice continuously differentiable, with prescribed boundary values

$u=u_0\text{on }\partial \Omega \mathrm{(}\mathrm{1}\mathrm{)}$

Write $u_{,\alpha }=\partial u/\partial x_{\alpha }$. Let $L(x,u,u_{,\alpha })$ have continuous second partials and define

$J[u]={\int }_{\Omega }L(x,u,u_{,\alpha })d^{m}x\mathrm{(}\mathrm{2}\mathrm{)}$

Variations $\eta \in C^2(\overline{\Omega })$ vanish on the whole boundary.

Derivation

From one independent variable to several.

Differentiating $J[u+\epsilon \eta ]$ at $\epsilon =0$,

$\delta J={\int }_{\Omega }[\frac{\partial L}{\partial u}\eta +\sum _{\alpha =1}^m\frac{\partial L}{\partial u_{,\alpha }}{\eta }_{,\alpha }]d^{m}x\mathrm{(}\mathrm{3}\mathrm{)}$

Integration by parts is now the product rule plus the divergence theorem. Write $A_{\alpha }=(\partial L/\partial u_{,\alpha })\eta$. Then

$\sum _{\alpha }\frac{\partial A_{\alpha }}{\partial x_{\alpha }}=\sum _{\alpha }\frac{\partial L}{\partial u_{,\alpha }}{\eta }_{,\alpha }+\sum _{\alpha }\frac{\partial }{\partial x_{\alpha }}\left(\frac{\partial L}{\partial u_{,\alpha }}\right)\eta \mathrm{(}\mathrm{4}\mathrm{)}$

Integrating (4) over $\Omega$ and applying the divergence theorem to the left side,

${\int }_{\partial \Omega }\sum _{\alpha }\frac{\partial L}{\partial u_{,\alpha }}\eta n_{\alpha }dS={\int }_{\Omega }\sum _{\alpha }\frac{\partial L}{\partial u_{,\alpha }}{\eta }_{,\alpha }{d}^{m}x+{\int }_{\Omega }\sum _{\alpha }\frac{\partial }{\partial x_{\alpha }}\left(\frac{\partial L}{\partial u_{,\alpha }}\right)\eta d^{m}x\mathrm{(}\mathrm{5}\mathrm{)}$

with $n_{\alpha }$ the outward unit normal. The surface integral vanishes because $\eta =0$ on $\partial \Omega$. Substituting the resulting expression into (3),

$\delta J={\int }_{\Omega }[\frac{\partial L}{\partial u}-\sum _{\alpha =1}^m\frac{\partial }{\partial x_{\alpha }}\left(\frac{\partial L}{\partial u_{,\alpha }}\right)]\eta d^{m}x\mathrm{(}\mathrm{6}\mathrm{)}$

By the multidimensional form of the fundamental lemma, given on the fundamental lemma, the bracket vanishes:

$\frac{\partial L}{\partial u}-\sum _{\alpha =1}^m\frac{\partial }{\partial x_{\alpha }}\left(\frac{\partial L}{\partial u_{,\alpha }}\right)=0\mathrm{(}\mathrm{7}\mathrm{)}$

$◼$

Equation (7) is the Euler-Lagrange equation for a field. It is a second-order partial differential equation. The structure is the same as the one-dimensional case throughout: differentiate under the integral, move the derivative off the variation, discard a boundary term, apply the lemma.

Several fields

With $N$ fields $u_1,\dots ,u_N$, independent variations give one equation per field, by the same select-one-index argument as on Several dependent variables:

$\frac{\partial L}{\partial u_i}-\sum _{\alpha =1}^m\frac{\partial }{\partial x_{\alpha }}\left(\frac{\partial L}{\partial u_{i,\alpha }}\right)=0,i=1,\dots ,N\mathrm{(}\mathrm{8}\mathrm{)}$

The boundary term does not always vanish

Two situations leave the surface integral in (5) standing, and both matter later.

Free boundary. If $u$ is unprescribed on part of $\partial \Omega$, the argument of Boundary terms and natural boundary conditions applies and yields the natural condition

$\sum _{\alpha }\frac{\partial L}{\partial u_{,\alpha }}{n}_{\alpha }=0\text{on that part}\mathrm{(}\mathrm{9}\mathrm{)}$

Second derivatives in $L$. If $L$ contains $u_{,\alpha \beta }$, two integrations by parts are needed and the surface term involves the normal derivative of $\eta$, which does not vanish merely because $\eta$ does. The Einstein-Hilbert action is exactly this case, and the standard remedy is to add a compensating surface term. That is taken up on the Einstein-Hilbert action.

Divergences and conservation

The combination appearing in (4),

$\mathrm{D}\mathrm{i}\mathrm{v}A=\sum _{\alpha =1}^m\frac{\partial A_{\alpha }}{\partial x_{\alpha }}\mathrm{(}\mathrm{10}\mathrm{)}$

is not an artefact of the derivation. It is the object Noether's theorems are about. Noether writes the identity produced by partial integration as

$\sum {\psi }_i\delta u_i=\delta f+\mathrm{D}\mathrm{i}\mathrm{v}A\mathrm{(}\mathrm{11}\mathrm{)}$

where the ${\psi }_i$ are the Lagrange expressions, that is, the left sides of (8).

Her first theorem states that invariance of $J$ under a $\rho$-parameter group makes $\rho$ independent combinations of the Lagrange expressions into divergences. Setting the field equations ${\psi }_i=0$ then gives $\mathrm{D}\mathrm{i}\mathrm{v}B=0$, and Noether notes that these divergence equations are what have lately been called laws of conservation.

In one independent variable a divergence is a total derivative and $\mathrm{D}\mathrm{i}\mathrm{v}B=0$ reduces to $B=\text{const}$, which recovers the first integrals of the Euler-Lagrange equation. Conservation laws in field theory and first integrals in mechanics are one result read in two dimensions of the independent variable. This is developed on Noether's first theorem.

Worked example: the vibrating string

Let $u(x,t)$ be the transverse displacement of a string of linear density $\rho$ under tension $\tau$. The Lagrangian density is kinetic minus potential energy per unit length:

$L=\frac{1}{2}\rho u_{,t}^2-\frac{1}{2}\tau u_{,x}^2\mathrm{(}\mathrm{12}\mathrm{)}$

$L$ contains no $u$, so $\partial L/\partial u=0$. The two derivative slots give $\partial L/\partial u_{,t}=\rho u_{,t}$ and $\partial L/\partial u_{,x}=-\tau u_{,x}$. Substituting into (7) with $(x_1,x_2)=(t,x)$,

$0-\frac{\partial }{\partial t}\left(\rho u_{,t}\right)-\frac{\partial }{\partial x}(-\tau u_{,x})=0\mathrm{(}\mathrm{13}\mathrm{)}$

that is

$\rho u_{,tt}=\tau u_{,xx}\mathrm{(}\mathrm{14}\mathrm{)}$

the wave equation, with speed $c=\sqrt{\tau /\rho }$. The full treatment, including a free end and the natural condition (9), is on Continuum systems: the vibrating string.

The machinery track is complete

Eight pages have carried the subject from the definition of a functional to a field equation. Everything after this point either constrains the machinery (Holonomic constraints), asks whether an extremal is actually a minimum (the second variation), restructures it (the Legendre transform), or applies it.

No page in this track has established that any extremal minimises anything. That is the correct state of the subject at this depth, and it is why this volume says stationary rather than least.

Next

Holonomic constraints opens the second mathematics track.