ME 5550 Intermediate Dynamics
Lagrange’s Equations for Multi-Degree-of-Freedom Systems
The configuration of systems with N degrees of freedom (DOF) can be defined in terms
of N generalized coordinates, say qk (k  1, , N ) . The differential equations of motion
(EOM) of the system can be derived using Lagrange's equations
d  K  K
 Fqk


dt  qk  qk
where
K

m
NB
1
2
j 1
j
R
vG j

2
(k  1, , N )

 R B j  H G j  kinetic energy of system
Fqk  generalized force associated with generalized coordinate qk
(due to all forces and torques acting on the system)
Note that it is important that K and Fqk be written only in terms of qk and qk and no other
variables.
If some of the forces and torques are conservative, then we can define their contribution
to the equation of motion in terms of a potential energy function. In this case, the differential
EOM can be derived from the equations
d  L 
L
 Fqk

 
dt  qk 
qk
 
nc
(k  1, , N )
where L  K  V is the Lagrangian of the system, V is the potential energy function for
the conservative forces and torques, and  Fqk  is the generalized force associated with qk
nc
for the nonconservative forces and torques, only.
Kamman – ME 5550 – page: 1/1