By Donald T. Greenwood
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127) i=1 We see that the total kinetic energy is the sum of three parts: (1) the kinetic energy due to the total mass moving at the speed of the reference point; (2) the kinetic energy due to motion relative to the reference point; and (3) the scalar product of the reference point velocity and the linear momentum of the system relative to the reference point. 127) is an important and useful result. It is particularly convenient in the analysis of systems having a reference point whose motion is known but which is not at the center of mass.
249) Note that Q x and Q y are the x and y components of F, whereas Q θ is the moment about particle 1. Principle of virtual work A system of particles is in static equilibrium if each particle of the system is in static equilibrium. A particle is in static equilibrium if it is motionless at the initial time t = 0, and if its acceleration remains zero for all t ≥ 0. Now consider a catastatic system of particles; that is, all transformation equations from inertial xs to qs do not contain time explicitly.
203) 37 Constraints and conﬁguration space or the alternate differential form n a ji (q, t) dqi + a jt (q, t) dt = 0 ( j = 1, . . 204) i=1 where, in either case, these expressions are not integrable. 200) would apply, indicating that the constraint is actually holonomic. 205), we ﬁnd that a ji ≡ ∂φ j , ∂qi a jt ≡ ∂φ j ∂t (i = 1, . . , n; j = 1, . . 206) for this holonomic constraint. 204). The holonomic case is distinguished by its integrability. Note that the coefﬁcients a ji (q, t) and a jt (q, t) are generally nonlinear in the qs and t.