This paper deals with mechanical systems subjected to a general class of non-ideal equality con- straints. It provides the explicit equations of motion for such systems when subjected to such nonideal, holonomic and/or nonholonomic, constraints. It bases Lagrangian dynamics on a new and more general principle, of which D'Alembert's principle then becomes a special case applicable only when the constraints become ideal. By expanding its perview, it allows Lagrangian dynamics to be directly applicable to many situations of practical importance where non-ideal constraints arise, such as when there is sliding Coulomb friction. One of the central problems in the field of mechanics is the determination of the equations of motion pertinent to con- strained systems. The problem dates at least as far back as Lagrange (1787), who devised the method of Lagrange mul- tipliers specifically to handle constrained motion. Realizing that this approach is suitable to problem-specific situations, the basic problem of constrained motion has since been worked on intensively by numerous scientists, including Volterra, Boltzmann, Hamel, Novozhilov, Whittaker, and Synge, to name a few. About 100 years after Lagrange, Gibbs (1879) and Appell (1899) independently devised what is today known as the Gibbs-Appell method for obtaining the equations of motion for constrained mechanical systems with nonintegrable equal- ity constraints. The method relies on a felicitous choice of quasicoordinates and, like the Lagrange multiplier method, is amenable to problem-specific situations. The Gibbs-Appell ap- proach relies on choosing certain quasicoordinates and elimi- nating others, thereby falling under the general category of elimination methods (Udwadia and Kalaba 1996). The central idea behind these elimination methods was again first devel- oped by Lagrange when he introduced the concept of gener- alized coordinates. Yet, despite their discovery more than a century ago, the Gibbs-Appell equations were considered by many, up until very recently, to be at the pinnacle of our under- standing of constrained motion; they have been referred to by Pars (1979) in his opus on analytical dynamics as ''probably the simplest and most comprehensive equations of motion so far discovered.'' Dirac considered Hamiltonian systems with constraints that were not explicitly dependent on time; he once more attacked the problem of determining the Lagrange multipliers of the Hamiltonian corresponding to the constrained dynamical sys- tem. By ingeniously extending the concept of Poisson brack- ets, he developed a method for determining these multipliers in a systematic manner through the repeated use of the con- sistency conditions (Dirac 1964; Sudarshan and Mukunda 1974). More recently, an explicit equation describing con- strained motion of both conservative and nonconservative dy- namical systems within the confines of classical mechanics was developed by Udwadia and Kalaba (1992). They used as their starting point Gauss's principle (1829) and considered general bilateral constraints that could be both nonlinear in the generalized velocities and displacements and explicitly depen- 1