Udwadia–Kalaba equation

In theoretical physics, the Udwadia–Kalaba equation is a method for deriving the equations of motion of a constrained mechanical system. This equation was discovered by Firdaus E. Udwadia and Robert E. Kalaba in 1992. The fundamental equation is the simplest and most comprehensive equation so far discovered for writing down the equations of motion of a constrained mechanical system. Although it is mostly a restatement of Newton's second law of motion, it makes a convenient distinction between externally applied forces and the internal forces of constraint, similar to the use of constraints in Lagrangian mechanics, but without the use of Lagrange multipliers. The Udwadia–Kalaba equation applies to a wide class of constraints, both holonomic constraints and nonholonomic ones, as long as they are linear with respect to the accelerations. The equation even generalizes to constraint forces that do not obey D'Alembert's principle.

In the study of the dynamics of mechanical systems, the configuration of a given system S is, in general, completely described by n generalized coordinates so that its generalized coordinate n-vector is given by

where T denotes matrix transpose. Using Newtonian or Lagrangian dynamics, the unconstrained equations of motion of the system S under study can be derived as a matrix equation (see matrix multiplication):

${\displaystyle \mathbf {M} (q,t){\ddot {\mathbf {q} }}(t)=\mathbf {Q} (q,{\dot {q}},t)\,,}$

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