2015 , Volume 20, № 5, p.53-64

In the present paper, we establish that the two-dimensional linear elasticity model admits a Hamiltonian structure. Moreover, the existence of extra conservation laws is also discussed. Notice that the main attention of the Hamiltonian theory of field systems is paid for nonlinear models and linearized models, which often inherit Hamiltonian structure of nonlinear equations. Nevertheless, the study of a Hamiltionian structure has sense when linearized equations are obtained, as a rule, without taking attention of the Hamiltonian structure for the original equations. So-called non-canonical form of the Poisson bracket of the nonlinear model under consideration can lead us to non-unique representations of linearized equations. We show that the two-dimension linear elasticity possesses a degenerated Poisson bracket and the integral of energy is a Hamiltonian. Based on the degenerate property of the obtained braсket, we found the complete family of the Casimir functionals, which depend on combinations of the second-order derivatives and these functionals are conserved for any form of Hamiltonian. The conditions of positive definiteness of the Hamiltonian which depend on parameters of the problem are obtained. We present connection between the obtained Hamiltonian structure and the form of equations rewritten in the Godunov symmetric representation. Using the Hamiltonian structure of the positive definite Hamiltonian, a skew-symmetric form for the quadratic Hamiltonian with the unit diagonal matrix is exposed. We prove the result that arbitrary linear symmetrical hyperbolic in the sense of Fridrichs is a non-canonical Hamiltonian system. The expression for the Poisson bracket and Hamiltonian using the coefficients of the corresponding symmetrical hyperbolic system is derived. Using the direct calculations, we find all zero-order conservation laws i.e. the conservation laws those densities are independent of the spatial derivatives. We prove that there no exist other zero-order conservation laws besides of the functional of energy.