Energy optimization, calculus of variations, Euler Lagrange equations in Maple

Alec Jacobson

August 16, 2016

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Here's a simple demonstration of how to solve an energy functional optimization symbolically using Maple.

Suppose we'd like to minimize the 1D Dirichlet energy over the unit line segment:

min  1/2 * f'(t)^2
 f
subject to: f(0) = 0, f(1) = 1

we know that the solution is given by solving the differential equation:

f''(t) = 0, f(0) = 0, f(1) = 1

and we know that solution to be

f(t) = t

How do we go about verifying this in Maple:

with(VariationalCalculus):
E := diff(f(t),t)^2:
L := EulerLagrange(E,t,f(t)):

so far this will output:

        L := {-2*diff(diff(x(t),t),t), -diff(x(t),t)^2 = K[2], 2*diff(x(t),t) = K[1]}

Finally solve with the boundary conditions using:

dsolve({L[1],f(0)=0,f(1)=1});

which will output

         t(t) = t