Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.

任何自由的幂等李代数均由其阶数和阶跃确定。我们考虑步骤3和4的自由幂零李李代数以及相应的连通和单纯连通Lie群。我们构造了此类基团的卡西米尔函数，即共轭表示的不变量。对于免费的三步幂等李群，我们获得了同伴轨道的完整描述。事实证明，一般共共轨轨道是仿射子空间，特殊共共轨轨道是仿射子空间或非奇异二次曲面的直接乘积。卡西米尔函数的知识对于研究动力学系统的积分性质和卡诺组上的最优控制问题很有用。特别是，对于三步自由卡诺组上的一些时间最优问题，