Worked Examples

One example per arc of the course, each chosen because it produces a result that would be genuinely hard to reach any other way.

• The catenary — the shape of a hanging chain, and why it is not a parabola
• Summing a series by residues — \(\sum 1/n^2 = \pi^2/6\) from a contour integral, with no series manipulation at all
• Water: symmetry without solving anything — how many vibrational modes H₂O has, what symmetry each one carries, and which are infrared active, deduced from the group alone

Related tools

The brachistochrone racer is the variational method's other classic; the contour integrator evaluates residues numerically for the second example's cousins.

The pattern to notice

In all three, a global constraint does the work: stationarity of a functional over all curves, analyticity over a whole region, invariance under a whole group. None of the three is solved by grinding through the local equations, and in the third case nothing is solved at all — the answer is forced by symmetry.