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36) The choice of generalized coordinates is not unique. , some of the local position coordinates (absolute coordinates) or differences between local coordinates (relative coordinates) may be chosen as generalized coordinates. 37) where y and y are the corresponding global position vectors. 31) may be rewritten for the whole system as r i (t) = ri (y, t) , S Ii (t) ≡ S i (t) = S i (y, t) , i = 1(1)p . 17) as ⎡ ⎡ ⎤ ⎤ 0 −∂si3 ∂si2 ∂si1 ⎣ ∂si3 0 −∂si1 ⎦ , ∂si = ⎣ ∂si2 ⎦ . 41) i := −∂si2 ∂si1 0 ∂si3 The 3 × f -functional or Jacobian matrices J T i , J Ri of translation and rotation, respectively, identify the relation between the local and the generalized or global coordinates.

Sγ α˙ cγ α˙ 0 −ω2 ω1 0 ⎡ ˙ ωR RZ = S RZ S ZR Then, it follows by rewriting the angular velocity vector ⎤ ⎡ ⎤ ⎡ ω1 cγ α˙ ⎣ ⎦ ⎣ ˙⎦ . ωR RZ = ω2 = sγ α γ˙ ω3 (2) (3) Comparing the computational efficiency of both approaches it turns out that the geometrical considerations (1) are more efficient. 3 Kinematics of a Rigid Body in a Moving Reference Frame From a mathematical point of view the kinematical description of the motion of a rigid body is most convenient in the inertial frame I resulting in a more simple representation of the fundamental laws of mechanics.

Rolling cylinder rolling motion without sliding on the plane results in contact lines between cylinder and plane which are parallel all the time. Thus, the center of mass C travels on a straight line, and the corresponding geometric constraint conditions read according to Fig. 15 as r1 ≡ 0 , r2 ≡ 0 , α ≡ 0, , γ≡0. (1) The original position and velocity variables {rC , S IK }, {v C , ωIK } are rewritten as (s= ˆ sin, c= ˆ cos) ⎤ ⎡ ⎤ ⎡ r1 cβ 0 −sβ (2) r IC = ⎣ 0 ⎦ , S IK = −β2 = ⎣ 0 1 0 ⎦ , sβ 0 cβ 0 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 r˙1 0 v IC = ⎣ 0 ⎦ , ω IIK = ⎣−β˙ ⎦ , ∂sIIK = ⎣−∂β ⎦ .