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9.7 Lagrange’s Equations

The discussion on energy methods focuses on deriving constitutive relations for energy-storing multi-
ports, and this can be very useful in some modeling exercises. For some cases where the constraint
relationships between elements are primarily holonomic, and definitely scleronomic (not an explicit
function of time), implicit multiport fields can be formulated (see Chapter 7 of [17]). The principal concern
arises because of dependent energy storage, and the methods presented can be a solution in some practical
cases. However, there are many mechanical systems in which geometric configuration complicates the
matter. In this section, Lagrange’s equations are introduced to facilitate analysis of those systems.
There are several ways to introduce, derive, and utilize the concepts and methods of Lagrange’s equations.
The summary presented below is provided in order to introduce fundamental concepts, and a thorough
derivation can be found either in Lanczos [20] or Goldstein [11]. A derivation using energy and power
flow is presented by Beaman, Paynter, and Longoria [3].
Lagrange’s equations are also important because they provide a unified way to model systems from
different energy domains, just like a bond graph approach. The use of scalar energy functions and minimal
geometric reasoning is preferred by some analysts. It is shown in the following that the particular benefits
of a Lagrange approach that make it especially useful for modeling mechanical systems enhance the bond
graph approach. A combined approach exploits the benefits of both methods, and provides a methodology
for treating complex mechatronic systems in a systematic fashion.

Classical Approach

A classical derivation of Lagrange’s equations evolves from the concept of virtual displacement and virtual
work developed for analyzing static systems (see Goldstein [11]). To begin with, the Lagrange equations
can be derived for dynamic systems by using Hamilton’s principle or D’Alembert’s principle.
For example, for a system of particles, Newton’s second law for the i mass, F i = p i , is rewritten, F i −
a ()
p i = 0. The forces are classified as either applied or constraint, F i = F i + f i . The principle of virtual work
is applied over the system, recognizing that constraint forces f i , do no work and will drop out. This leads
to the D’Alembert principle [11],

∑ ( F i – p ˙ i) δr i = 0 (9.28)
a ()
⋅
i
The main point in presenting this relation is to show that: (a) the constraint forces do not appear in this
formulative equation and (b) the need arises for transforming relationships between, in this case, the N
coordinates of the particles, r i , and a set of n generalized coordinates, q i , which are independent of each
other (for holonomic constraints), i.e.,

r = r (q , q ,…,q , t) (9.29)
i
n
2
1
i
By transforming to generalized coordinates, D’Alembert’s principle becomes [11]
 ∂T 


∑  ----- ------- – -------  Q j δq j = 0 (9.30)
d ∂T
–


j  dt ∂q ˙ j ∂q j 
where T is the system kinetic energy, and the Q j are components of the generalized forces given by

Q j ∑ F i ⋅ ∂r i
=
-------
∂q j
i
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