By Donald T. Greenwood
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Another possibility might be to use the cylindrical coordinates φ and z as qs, where φ effectively gives the longitude and z the latitude. Nonholonomic constraints Nonholonomic constraints may have the general form ˙ t) = 0 f j (q, q, ( j = 1, . . 202) but usually they have a simpler form which is linear in the velocities. Thus, we nearly always assume that nonholonomic constraints have the form n fj = a ji (q, t) q˙ i + a jt (q, t) = 0 i=1 ( j = 1, . . 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, .
95) i=1 where ρi is now the position vector of particle m i relative to the center of mass. Now let us write an expression for Hc when P is not at the center of mass. 98) This important result states that the angular momentum about an arbitrary point P is equal to the angular momentum about the center of mass plus the angular momentum due to the relative translational velocity ρ˙c of the center of mass. Of course, this result also applies to the case of a ﬁxed reference point P when ρ˙c is an absolute velocity.
14. First consider the force acting on particle C due to particles A and B. 162) 2 l0 3 r0 √ since l0 = 3 r0 . 164) Now let us assume the initial conditions r (0) = r0 , r˙ (0) = 0, ω(0) = 12 ω0 . We can use conservation of energy and of angular momentum to solve for the minimum value of r , and therefore of l. Let us concentrate on particle C alone since the system values are three times those of a single particle. 165) which can be integrated to yield the general expression Gm 2 V = −√ 3r where the integration constant is set equal to zero.