Equipartition of energy for a class of second order equations

Abstract

We consider the Cauchy problem for a class of second order equations of the form ( d / d t − A 2 ) ( d / d t − A 1 ) u ( t ) = 0 (d/dt - {A_2})(d/dt - {A_1})u(t) = 0 in a Hilbert space H. A d’Alembert type solution formula is presented and we give a suitable definition of energy. Also, we derive a necessary and sufficient condition for the asymptotic equipartition of energy (Kinetic and Potential) to hold. These results generalize corresponding results for the abstract wave equation ( d 2 / d t 2 + A 2 ) u ( t ) = 0 ({d^2}/d{t^2} + {A^2})u(t) = 0 .