A mixed system of six equations of elasticity is represented as a Hamiltonian (canonical) operator system in one of the spatial coordinates. It is shown that this system is the Euler equations for the Hellinger-Reissner principle with an appropriately modified integrand. One more functional with an operator integrand from which the canonical operator system can be derived is set up Keywords: mixed system of equations of elasticity, Hamiltonian (canonical) operator system, Hellinger-Reissner principle, variational principle with an operator integrand, Euler equationsIn the classical theory of elasticity, the system of three equilibrium equations, six constitutive equations, and the Cauchy relations for strains are reduced to three Lame equations for displacements. The formulation of the problem for stresses uses three equilibrium equations and six Beltrami-Michell equations derived from Saint-Venant's strain compatibility equations and strain-stress relationship for an isotropic body, only three of them being independent. A mixed system of six equations for three displacements and three stresses is also used [2-5, etc.], especially in combination with numerical methods in one spatial coordinate. These equations are written in Cauchy normal operator form for the first derivatives with respect to the same coordinate. The monograph [8] was the first to show that it is possible to represent the mixed system in Hamiltonian (canonical) operator form in a spatial coordinate. This representation is of fundamental value in the theory of vibrations and waves in periodically inhomogeneous media [2, 8, 9, 11, etc.]

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We carried out a general analysis of the mixed systems of six equations of a Timoshenko-type theory of vibrations of plates in rectangular and polar coordinates. It is shown that these systems can be represented in the operator canonical (Hamiltonian) form with respect to the spatial coordinate under the appropriate choice of "canonical" variables and the operator Hamilton function. Some functionals for canonical systems are constructed. For a mixed Hamiltonian system, a reduced Hellinger-Reissner functional of the spatial coordinate in the rectangular coordinates is obtained. The variation of the functional yields six canonical equations.