Branching of the solutions and polynomial integrals of the equations of dynamics. (Q2714631)

The author treats the equations of motion for natural mechanical systems \(\dot x_i = \frac{\partial K}{\partial y_i},\quad \dot y_i = -\frac{\partial K}{\partial x_i} + F_i;\quad 1\leq i\leq n,\) where \(K = \frac12\,\sum\limits_{i,j=1}^n g_{ij}(x)y_iy_j. \) Obtained are branching conditions for the solutions to system (1) in the plane of complex variable. A relationship between the solution branchings and the number of independent integrals that are polynomial with respect to impulses is discussed. The application of general results is shown by examples from dynamics.