Hamilton’s principle applied to piezomagnetism and related variational principles

Abstract

In piezomagnetism, the fundamental equations have been developed in differential form [e.g., V. I. Alshits and A. N. Darinskii, Wave Motion 15, 265–283 (1992)]. Alternatively, they may be expressed in variational form with its well-known features; this is the topic of this paper. First, the magnetic vector, that is, the gradient of the magnetic potential, is introduced [cf. the authors, Int. J. Solids Struct. 40, 4699–4706 (2003)]. Second, the sufficient conditions based on the energy argument are enumerated for a unique solution in the fundamental equations. Third, Hamilton’s principle is stated and a three-field variational principle is obtained. The principle yields only the divergence equations and some natural boundary conditions, and it has the remaining fundamental equations as its constraint conditions. The conditions are generally undesirable in computation, and they are accordingly removed through an involutory transformation [e.g., the authors, Int. J. Eng. Sci. 40, 457−489 (2002)]. Thus, a unified variational principle operating on all the field variables is derived in piezomagnetism. The principle is shown, as special cases, to recover some of earlier ones. [Work supported by TUBA.]