Abstract
A description of all of the cases of the existence of a linear invariant relation of Kirchhoff's equations is obtained. Existence conditions in a parametric vector form and in an explicit coordinate form are presented. New linear invariant relations and partial solutions are found. The inclusion of well-known linear invariant relations in the general case obtained is indicated. A geometrical interpretation of the conditions for the Chaplygin linear invariant relation to exist is given.
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