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0593_C12_fm Page 423 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 423

or

[
˙ ˙
θ
θ
mgrsinθ + mr −( ) +( ) 2 ψ φ cosθ +( ) φ ˙ sin cosθ = 0 (12.2.44)
˙˙
2
54
3
54
θ
˙˙ ψ
˙ ˙
φ
˙˙
2
0 + mr 2 −( [ 3 2) sinθ −( 3 2) sin θ −( 5 2)φθ sin cosθ
(12.2.45)
−( 14) cosφ 2 θ +( 1 2)ψθ cosθ ] = 0
˙˙
˙ ˙
φ
˙ ˙
˙˙
˙˙
0 − mr 2 ( [ 3 2)ψ +( 3 2) sinθ +( 5 2)θφ cosθ ] = 0 (12.2.46)
After simplification, these equations may be written in the forms:
( 4gr)sinθ − θ 6 ˙ ˙ cosθ + 5φ 2 ˙ sin cosθ = 0 (12.2.47)
5 + ψ φ
˙˙
θ
φ cosθ + 2 ψθ = 0 (12.2.48)
˙˙
˙ ˙
˙ ˙
φ
˙˙
3 ˙˙ ψ + 3 sinθ + 5φθ cosθ = 0 (12.2.49)
(Note that in this simplification process the terms of Eqs. (12.2.46) were recognized as the
first three terms in Eq. (12.2.45).)
Equations (12.2.47), (12.2.48), and (12.2.49) are identical to Eqs. (8.13.16), (8.13.17), and
(8.13.18), obtained using d’Alembert’s principle.

12.3 Lagrange’s Equations

Next to Newton’s laws, Lagrange’s equations are probably the most widely used equations
for studying systems with several degrees of freedom. Lagrange’s equations can be
obtained directly from Kane’s equations. Unlike Kane’s equations, however, Lagrange’s
equations are primarily restricted to holonomic systems.
Consider a holonomic mechanical system S with n degrees of freedom represented by
the coordinates q (r = 1,…, n). Then, the generalized inertia forces F * may be expressed
r r
in terms of the kinetic energy K of S as [see Eq. 11.12.5)]:

d  ∂ K  ∂ K
F =− + ( r =… ) (12.3.1)
*
n ,
, 1


r dt ∂   q ∂ r
q ˙
r
Kane’s equations state that (see Eq. (12.2.1)):
F + F = 0 ( r = …, n) (12.3.2)
*
1
,
r r