On steady-state motions of a mechanical conservative autonomous systems

This article proposed a justification for the Routh-Lyapunov method for finding the stationary motions of autonomous conservative mechanical systems. The presentation was initially conducted for autonomous Hamiltonian systems, on which additional conditions in the form of certain algebraic equalities were imposed. Their application is carried out by the well-known Lagrange method of implicitly taking into account the indicated conditions, which consists of forming a bundle of the Hamilto- nian of the initial system and the algebraic sum of the specified conditions, multiplied by undefined real factors. This method consists of forming a connection between the Hamiltonian of the original system and the algebraic sum of the specified conditions, multiplied by undefined real factors. The latter are subsequently considered on an equal basis with the system variables. Using a new view of the integration of Hamiltonian systems as an infinitesimal contact transformation, the latter essentially reduced to finding the equilibrium positions of the transformed system, taking into account the introduced constraints. Taking into consideration the fact that in conservative systems the Hamiltonian consists of the sum of kinetic and potential energies, this approach is precisely trans- ferred to mechanical autonomous conservative systems of general form. For this purpose, a bundle is formed consisting of the algebraic sum of the first integrals of the original system of equations of motion with indefinite real multipliers participating on an equal basis with the phase variables. The condition of constancy of solutions for stationary motion reduces the problem of finding them to the vanishing of all partial derivatives with respect to phase vari- ables and Lagrange multipliers both in Hamiltonian systems and in mechanical autonomous conservative systems of general form. In the general case, a discrepancy was established between the found by the Routh- Lyapunov method of the stationary motions and the solutions of the equilibrium posi- tions of the studied system.