Classical motion in isospectral billiards

Isospectral domains are non-isometric enclosures possessing identical energy spectra. Semiclassical reasoning suggests that classical billiards shaped like isospectral domains will possess the property of iso-length-spectrality. Namely, for every periodic orbit in one of the billiards, there will be an orbit of identical geometric length in the other billiard. This result can also be proved using only classical considerations [1]. Here it is demonstrated for both pseudointegrable and chaotic cases, using mathematics easily accessible to an undergraduate physicist.