SYNTHESIS OF HUMAN MOVEMENT—FORWARD SOLUTIONS - Biomechanics and Motor Control of Human Movement - page 207

8.2.3 The Lagrangian Function L
The Lagrangian function L is defined as the difference between the total
kinetic energy KE and the total potential energy PE in the system,

L =

−

KE PE (8.5)

The kinetic energy for a segment is defined as the work done on the
segment to increase its velocity from rest to some value v , where v is mea-
sured relative to a global (inertial) reference system. The existence of an
inertial reference system is a fundamental postulate of classical dynamics.
Potential energy exists if the system is under the influence of conservative
forces. Hence, segment potential energy is defined as the energy possessed
by virtue of a segment (or particle) position in a gravity field relative to
a selected datum level (usually ground level) in the system. In case of a
spring, potential energy is the energy stored in the spring because of its elastic
deformation.

8.2.4 Generalized Forces [ Q ]
A nonconservative force F j acting on a segment can be resolved into compo-
nents corresponding to each generalized coordinate (q i , i 1, ... , n) in the

system. This is also true for constraint forces. A generalized force Q i is the
component of the forces that do work when q i is varied and all other gen-
eralized coordinates are kept constant. In more useful terms, if f forces are
acting on the system, then:

=

λ j F xj ∂R xj
∂q i

f

F yj ∂R yj
∂q i

F zj ∂R zj
∂q i

Q i = + + (8.6)

j = 1

where R j is the position vector of the force F j . Moments that are generated by
these forces or externally applied moments are greatly affected by the choice
of the angular system. More on this topic later. In the case of m constraint
equations,

m

λ j ∂ j
∂q i

Q i = (8.7)

j = 1

Equations (8.6) and (8.7) are added together before they are used.