Symplectic Elasticity: Theory and Applications

Abstract

Many of the early works on symplectic elasticity were published in Chinese and as a result, the early works have been unavailable and unknown to researchers worldwide. It is the main objective of this paper to highlight the contributions of researchers from this part of the world and to disseminate the technical knowledge and innovation of the symplectic approach in analytic elasticity and applied engineering mechanics. This paper begins with the history and background of the symplectic approach in theoretical physics and classical mechanics and subsequently discusses the many numerical and analytical works and papers in symplectic elasticity. This paper ends with a brief introduction of the symplectic methodology. A total of more than 150 technical papers since the middle of 1980s have been collected and discussed according to various criteria. In general, the symplectic elasticity approach is a new concept and solution methodology in elasticity and applied mechanics based on the Hamiltonian principle with Legendre’s transformation. The superiority of this symplectic approach with respect to the classical approach is at least threefold: (i) it alters the classical practice and solution technique using the semi-inverse approach with trial functions such as those of Navier, Lévy, and Timoshenko; (ii) it consolidates the many seemingly scattered and unrelated solutions of rigid body movement and elastic deformation by mapping with a series of zero and nonzero eigenvalues and their associated eigenvectors; and (iii) the Saint–Venant problems for plane elasticity and elastic cylinders can be described in a new system of equations and solved. A unique feature of this method is that bending of plate becomes an eigenvalue problem and vibration becomes a multiple eigenvalue problem.