This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems, where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case by applying the Pontryagin maximum principle. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws, a minimum required value of the coefficient of sliding friction is defined, so that the considered system is moving in accordance with nonholonomic bilateral constraints

The paper brings forth a detailed analysis of the solution of the problem of the material realization of a nonlinear nonholonomic constraint (NNC). The existing models of the NNC are shown that can be classified into two groups: the first group comprises correctly realized physical models, while the second group contains the so‐called “quasinonlinear” nonholonomic constraints which in fact represent mathematical models. The correctness of the cited models is considered in detail, and the essential nature of such constraints, the basic of which is holonomic, is shown. The second group of models, i.e., the “quasinonlinear” NC (nonholonomic constraints) in fact represents the given program of motion, while the additional force, which carries out the program, has the analytical form of the reaction of the NNC. That is why are presented the models of the NNC which possess a clear physical sense, on the basis of which certain statements on the method of variation and the reaction of the NNC can be given. With regard to the clear physical sense and the nature of the models cited, the NNC that come out of them are used quite normally in the analysis of motion of such a system. The cited models, together with standard models oh nonholonomic Mechanics (sphere, disk, blade) make a group of basic nonholonomic constraints which can be classified, according to the three criteria, into certain types. Finally, it is shown that the cited model can be used for the construction of “nonholonomic chains”, both open and closed ones.

Depending on how the nonholonomic constraints have been introduced to the Lagrange‐D'Alemberts's principle, there are several differential equations of motion in the mechanics of nonholonomic systems. In this work, the most general type of differential equations of motion (fundamental to all known forms of the equations of motion for nonholonomic as well as holonomic systems) is derived. Here, the equations represent the generalization of Poincare's equation [1]. In published works [2, 3, 4, 5, 6], these have already taken into account nonlinear nonholonomic constraints and linear relations between real velocities and kinematic parameters. A method of dedication of the most generalized form of the equations of motion will be shown. It is followed by the analysis of particular cases. Then, it will be shown how to get form the generalized form to Maggi, Appell, Voronec, Chaplygin, Volterra, Ferrers, and Boltzmann‐Hamel's equations appearing in nonholonomic systems. Further, a system of material points of variable mass, where the equations of motion are derived for the most general case of reactive forces and in case of constraints depending on mass variables will be considered. All theoretical considerations are illustrated with the analysis of the relevant nonholonomic model.

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The paper analyzes the motion of nonholonomic mechanical systems composed of two particles with imposition of various nonlinear limitations to the velocities of the particles – parallelism of velocities, equality of the intensities of velocities and perpendicularity of velocities. The analysis for such systems includes: equations of constraints, reactions of constraints, i.e. the mode of variations of such constraints, trajectories of the points of the systems, linear integrals for generalized velocities, i.e. cyclical coordinates. It is clearly demonstrated on these models that in the case of nonlinear nonholonomic constraints the Hamiltonian effect, in the general case, has no stationary value. Lastly, the equations of brachistochronic motion of described systems are derived and the brachitochrones of specified points are determined.