An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.