Traditionally,analytical mechanics is described by means of dynamics,and the common foundation for structural mechanics and optimal control is analytical mechanics.So,under the framework of structural mechanics or optimal control theory,there should also be a whole set of analytical mechanics theory,which we define as analytical structural mechanics.A conservative system can be described with the Hamilton system methodology,and its characteristic is the symplectic conservation,which is the most important feature of conservative system.The finite element method was initiated from structural mechanics,and its element stiffness matrices should be symmetric,which is,in fact,the symplectic conservation.Based on the fact that the interval deformation energy depends only on the two end displacements vector,we derive the Lagrange and Poisson brackets analytically,the symplectic duality system,the canonical equations,and the canonical transformations,etc.