ABSTRACT
Throughout this text, we’ve seen how energy minimization principles play a key role in understanding self-assembly. Often, the energy for a system can be expressed in terms of an energy integral. In this case, several techniques are available for finding the extremals. One of the most important is the indirect method wherein the Euler-Lagrange equation is used to essentially turn the problem into a problem about differential equations. Here, we give an informal derivation of the Euler-Lagrange equation for integrals of the type:
I = ∫ x2 x1
F (x, y, y′) dx. (A.1)
The basic idea is to turn the problem of minimizing the integral over a set of functions into the problem of minimizing a function of a single real variable. Performing a minimization of a function of a single real variable is then a simple calculus exercise. To accomplish this, we first imagine that we know the actual minimizer y(x). Assume that y(x1) = y1 and y(x2) = y2. Consider the function
y(x) + :η(x), (A.2)
where we impose the conditions η(x1) = η(x2) = 0. This function is : away from the exact solution y(x) and agrees with the exact solution at the boundary points x1 and x2. See Figure A.1. Plug this function into the integral to be minimized to obtain
I(:) = ∫ x2 x1
F (x, y + :η, y′ + :η′) dx. (A.3)
Now, notice that I(:) is a function of the single real variable : and that I(0) must be a minimum since y(x) was assumed to be the actual minimizer for I! But, I(:) viewed as a function of the real variable : having a minimum at : = 0 implies that
dI
d: (0) = 0. (A.4)
So, let us compute the derivative of equation (A.3) with respect to epsilon, evaluate at : = 0 and set the result equal to zero as required by equation (A.4). We note that
∂F
∂:
∣∣∣∣ =0
= ∂F
∂y η +
∂F
∂y′ η′. (A.5)
FIGURE A.1: A variation from the actual minimizer of the energy. The minimizer, y(x), and the variation, y(x) + :η(x), agree at the endpoints.
