m-vector. We note that this set of constraint equations encompass a very general variety of holonomic and non-holonomic equality constraints. For example, holonomic constraints of the form can be differentiated twice with respect to time while non-holonomic constraints of the form can be differentiated once with respect to time to obtain the m-by-n matrix A and the m-vector b. In short, constraints may be specified that are nonlinear functions of displacement and velocity, explicitly dependent on time, and functionally dependent. As a consequence of subjecting these constraints to the unconstrained system S, an additional force is conceptualized to arise, namely, the force of constraint. Therefore, the constrained system Sc becomes where Qc—the constraint force—is the additional force needed to satisfy the imposed constraints. The central problem of constrained motion is now stated as follows: given the unconstrained equations of motion of the system S, given the generalized displacement q(t) and the generalized velocity of the constrained system Sc at time t, and given the constraints in the form as stated above, find the equations of motion for the constrained system—the acceleration—at time t, which is in accordance with the agreed upon principles of analytical dynamics. Notation Below, for positive definite , denotes the inverse of its square root, defined as , where is the orthogonal matrix arising from eigendecomposition (whose rows consist of suitably selected eigenvectors of ), and is the diagonal matrix whose diagonal elements are the inverse square roots of the eigenvalues corresponding to the eigenvectors in . Equation of motion The solution to this central problem is given by the Udwadia–Kalaba equation. When the matrix M is positive definite, the equation of motion of the constrained system Sc, at each instant of time, is where the '+' symbol denotes the pseudoinverse of the matrix . The force of constraint is thus given explicitly as and since the matrix M is positive definite the generalized acceleration of the constrained system Sc is determined explicitly by In the case that the matrix M is semi-positive definite , the above equation cannot be used directly because M may be singular. Furthermore, the generalized accelerations may not be unique unless the (n + m)-by-n matrix has full rank (rank = n). But since the observed accelerations of mechanical systems in nature are always unique, this rank condition is a necessary and sufficient condition for obtaining the uniquely defined generalized accelerations of