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The trend of the modern world is aimed at improving the energy efficiency of buildings and reducing unnecessary costs of heat and electricity. In view of this, heating systems are used with a periodic operating time, in order to reduce the cost of heat energy during periods of absence of people. As a result, this paper will consider a method for determining the temperature of a heating device, taking into account the heat accumulation by the metal body of the device and the heat carrier inside it, with changing environmental parameters, which will allow taking into account the accumulated heat during the heating system operation.

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In this paper, on the basis of the variational principle of the least Hamilton action, the field-theoretic Lagrange equations of the 2nd kind are derived in the presence of a dependence of the density of the Lagrange function on the higher derivatives. On the basis of the obtained equations, written in terms of variables that most fully reflect hypotheses and used assumptions, the equations of continuum mechanics are obtained and the Timoshenko, Rayleigh and Euler-Bernoulli beam models are calculated. It is shown that the method used leads to a more efficient method of obtaining partial differential equations describing the dynamics of deflection of beams. The possibility of taking into account the distributed load and the elastic base in the Lagrangian formalism is noted. The possibility of further development of analytical and theoretical-physical methods in application to mathematical modeling of building structures is analyzed.

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