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Dynamical problems subject to non-linear non-holonomic constraints are analyzed using the differential variational principles of Jourdain (JVP) and Gauss (GVP). In contrast with linear non-holonomic constraints which arise from passive contact or rolling constraints, non-linear non-holonomic constraints (NNHC) arise in problems where the constraints are actively imposed, such as control or servo-constraints, consideration of some first integrals as kinetic side conditions, or mathematical constraints without regard to their physical realizability. To date, such problems have been treated by various ad hoc and unrelated methods in a fragmented manner, without a unifying idea running through the various methods of approach. In the present paper, it is shown that by applying the JVP to velocity constraints, and the GVP to acceleration constraints, dynamical problems subject to NNHC can be handled in a uniform way. Both the method of constraint embedding (elimination of variations) and constraint adjoining (use of Lagrange multipliers) are used in analyzing the behavior of the dynamical system subject to NNHC.

The Appell-Hamel non-holonomic problem which is intrinsically linear, has been formulated by the same authors as a non-linear problem by means of a limiting procedure. This limiting process has been shown to be incorrect, in addition to being unnecessary in problems with rolling-type contact, where the constraint relations are known to be linear. It is well established however, that there exist non-linear non-holonomic constraints that arise from sources other than rolling-type contact. In order to be able to study the effect of constraints that arise from sources other than rolling, the Appell-Hamel problem is reformulated in terms of non-linear non-holonomic constraints, leading to the same results as the linear formulation, thus validating the non-linear analysis. A complete set of constants of the motion are derived, and limiting motions are investigated. The paper concludes with a description of the general method of solution to determine the time-history of the generalized coordinates. It ends with a detailed discussion of the numerical integration of the equations of motion.

A non-holonomic problem proposed by Greenwood is solved by reducing it to a Caratheodory problem. The resulting equations of motion are formulated in terms of natural physical variables corresponding to quasi-velocities. A connection between the two available methods of integration of the equations of motion is established that closes an existing gap in the literature. The explicit analytical results obtained for the particular set of initial conditions are confirmed by the numerical integration of the equations of motion performed on a computer. Thus, the procedure developed can be easily extended to a different set of input data. The time histories and trajectories obtained on the computer also suggest the development of asymptotic solutions for short and long times. Such results are obtained analytically and displayed in closed form.

An ad hoc procedure is given for obtaining first integrals of second order differential equations in which the non-linear term is a power of the dependent variable, as in the Emden-Fowler equation. The main theorem is a considerable extension of previous results along these lines. A corollary implies several known examples.

A direct, constructive approach to the problem of finding first integrals of certain non-linear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries] in the context of Noether's theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a two-dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence, a first integral which is independent of the energy integral exists for a particular Hamiltonian of the Contopoulos type.

The direct method for the construction of first integrals, developed in a previous paper, is applied to a system of second-order ordinary differential equations with time-varying coefficients. The connection between the existence of a Lagrangian and the problem of finding quadratic integrals, surmised in previous papers because of specific results, is rigorously established. Finally, the approach is illustrated by obtaining all the known quadratic integrals for a time-dependent n -dimensional harmonic oscillator from a single expression.

A method based on a differential variational principle is developed in order to extend Noether's theorem to constrained non-conservative dynamical systems. The result is applied to generate constants of the motion for a generic example of a non-linear, dissipative dynamical system with time-varying coefficients represented by the Emden equation. The converse of Noether's theorem, whereby the symmetries of the system are determined from the knowledge of the Lagrangian and a first integral is also considered for both the Emden equation, and that of the damped harmonic oscillator. It is further shown that the presence of ideal constraints (whether holonomic or non-holonomic) does not affect the statement of Noether's theorem. The constraints affect the Jacobi energy integral, however, because they enter into consideration through real work instead of virtual work. It is shown that the Jacobi integral is conserved provided that: (a) the Lagrangian is explicitly independent of time, (b) the real power of the generalized forces not derivable from a potential vanish, (c) the holonomic constraints are explicitly independent of time, (d) the non-holonomic constraints are linear and homogeneous in the generalized velocities.