d ∂R =d (∂⅋ + βr) =d ∂⅋ +∂βr q˙1 +∂βr q˙2+ ...,
dt ∂q˙rdt ∂q˙rdt ∂q˙r∂q1 ∂q2

(14)

∂R =∂⅋ −∂K +∂β1 q˙1 +∂β2 q˙2 + ....
∂qr ∂qr∂qr ∂qr∂qr

(15)

Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form

d ∂⅋ −∂⅋ + (r, 1) q˙1 + (r, 2) q˙2 + ... + (r, s) q˙s + ... +∂K = Qr,
dt ∂q˙r∂qr ∂qr

(16)

where

(r, s) = ∂βr− ∂βs.
∂qs ∂qr

(17)