Construction of Linear Invariant Relations of Kirchhoff Equations

Abstract

For the Kirchhoff equations in the general case, when the cross-term matrix of the Hamiltonian can be asymmetric, the existence conditions for a linear invariant relation of a general form that connect the impulsive moment and the impulsive force are obtained. It is shown that equations with such an invariant relation can be transformed into the equations with an invariant relation for an impulsive moment, the existence conditions of which with a symmetric cross-term matrix coincide with the Chaplygin case. A description of a set of Hamiltonians that admit the existence of a linear invariant relation of a general form is given. The coordinate form of the invariant relation and its existence conditions is obtained. The total number of conditions is six (in contrast to the eight conditions in the Chaplygin case). Reduction to the Riccati equation is preformed. It is shown that if there is a linear integral, then the Kirchhoff equations are reduced to the Kirchhoff case.