the constrained system Sc at each instant of time. Thus, when has full rank, the equations of motion of the constrained system Sc at each instant of time are uniquely determined by (1) creating the auxiliary unconstrained system and by (2) applying the fundamental equation of constrained motion to this auxiliary unconstrained system so that the auxiliary constrained equations of motion are explicitly given by Moreover, when the matrix has full rank, the matrix is always positive definite. This yields, explicitly, the generalized accelerations of the constrained system Sc as This equation is valid when the matrix M is either positive definite or positive semi-definite. Additionally, the force of constraint that causes the constrained system Sc—a system that may have a singular mass matrix M—to satisfy the imposed constraints is explicitly given by Non-ideal constraints At any time during the motion we may consider perturbing the system by a virtual displacement δr consistent with the constraints of the system. The displacement is allowed to be either reversible or irreversible. If the displacement is irreversible, then it performs virtual work. We may write the virtual work of the displacement as The vector describes the non-ideality of the virtual work and may be related, for example, to friction or drag forces (such forces have velocity dependence). This is a generalized D'Alembert's principle, where the usual form of the principle has vanishing virtual work with . The Udwadia–Kalaba equation is modified by an additional non-ideal constraint term to Examples Inverse Kepler problem The method can solve the inverse Kepler problem of determining the force law that corresponds to the orbits that are conic sections. We take there to be no external forces (not even gravity) and instead constrain the particle motion to follow orbits of the form where , is the eccentricity, and is the semi-latus rectum. Differentiating twice with respect to time and rearranging slightly gives a constraint We assume the body has a simple, constant mass. We also assume that angular momentum about the focus is conserved as with time derivative We can combine these two constraints into the matrix equation The constraint matrix has inverse The force of constraint is therefore the expected, central inverse square law Inclined plane with friction Consider a small block of constant mass on an inclined plane at an angle above horizontal. The constraint that the block lie on the plane can be written as