Constraints

Frequently a minimization problem includes a constraint. In ordinary multivariable calculus, for instance, one might be asked to find the minimum of f(x,y,z) under the constraint that g(x,y,z)=0. This is usually done by the method of Lagrange multipliers.

Example.

To minimize with the constraint , we introduce a Lagrange multiplier and define . Now we minimize h in the usual way:

These two equations, together with the constraint , give three equations for the three unknowns x, y, and  .

An explanation of why the method works can be found in most first-year textbooks.

If there is more than one constraint, we use more than one Lagrange multiplier, one multiplier for each constraint.

Example.

To minimize with the constraints , , we introduce Lagrange multipliers and and define . Setting three partial derivatives to zero, along with the two constraints, gives five algebraic equations for the five unknowns x, y, z, , .

Constrained minimization of a functional is no different. To minimize S[y(x)] under the constraint T[y]=0, define the combined functional . Then write the minimization equation

 

This will yield an equation for y(x) as always, with appearing as a parameter. The constraint equation T[y]=0 gives another equation, so that there is enough to fix , too.

Example.

Let's minimize , subject to the boundary conditions y(0)=y(1)=0 and to the constraint . Then , and the minimization (54) gives the Euler equation

This has the general solution . Plugging this into the constraint gives . Solving this equation together with the boundary conditions gives , , and , so the solution is y=6x(1-x).

If there is more than one constraint, say and , then we use two Lagrange multipliers and we minimize .

A more complex problem is that of a locally constrained minimization. Here there is typically a functional of (at least) two functions, S[x(t),y(t)], along with an equation that constrains x(t) and y(t) at every time t. Since there are an infinite number of constraints--one at every time t--there must be an infinite number of Lagrange multipliers--one at every time t. So we have a Lagrange multiplier function . If the constraint is f(x(t),y(t))=0 then we define the new functional

When we minimize R with respect to x(t) and y(t), we get differential equations in which the unknown function appears. In addition, the algebraic equation f(x(t),y(t))=0 is available to help fix .

Example.

Consider a particle moving in a plane and constrained to move along the curve . The action is

and the constraint is . We define

the minimization of which gives the Euler-Lagrange equations

Since in these differential equations is an unknown function, their solution is quite difficult.

The example shows that the Lagrange multiplier method for local constraints is not really a calculational tool. It is, however, very important as a conceptual device in the analytical mechanics of constrained systems, and reappears in classical and quantum field theory.