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On a Lie group, we introduce a family of group-invariant metrics and show that the curves invariant under this group are spirals in all the introduced metrics (i.e., they have constant curvatures). An important role is played by an algebra, which we call dual, defined on the same group. The main relation between these algebras is that the trajectories of the one-parameter groups generated by one algebra are invariant curves in the metric that is invariant under the other algebra. The fact that these curves are spirals distinguishes our approach from that of Cartan, who considered the trajectories of one-parameter groups as geodesics in some metric. The presented results are related to the analysis of the geometric meaning of the previously obtained classification of one-dimensional kinetic equations, where invariant curves are the trajectories of particles. [ABSTRACT FROM AUTHOR]

We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift group does not admit uniform dilations, because the left- and right-hand sides of the equation scale differently. In particular, this holds for the classical Boltzmann equation. For the Vlasov equation, we also obtain the equivalence group and classifying relations for the symmetry group and classify the interparticle interactions for which the Vlasov equation admits groups containing the Galilean group in the case where external forces are absent. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional Boltzmann equation f t + c f x + (F f) c = 0 with a function F depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and d c = F d t , (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them. [ABSTRACT FROM AUTHOR]

We present the group classification of the one-dimensional Boltzmann equation with respect to the function F = F(t, x, c) characterizing an external force field under the assumption that the physically meaningful constraints dx = c dt, dc = F dt, dt = 0, and dx =0 are imposed on the variables. We show that for all functions F, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero F. [ABSTRACT FROM AUTHOR]