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242 Dynamics of Mechanical Systems

Recently, researchers have established that Newton’s first law was known and stated in
China in the third or fourth century BC. Under the leadership of Mo Tzu it was stated [8.3]:

The cessation of motion is due to the opposing force. ...If there is no opposing force ...
the motion will never stop. This is as true as that an ox is not a horse.

Newton’s laws form the foundation for the principles of dynamics employed in modern
analyses. We will briefly review some of these principles in the following section. We will
then focus upon d’Alembert’s principle in the remaining sections of the chapter and will
illustrate use of the principle with several examples. We will consider other principles in
subsequent chapters.

8.2 Principles of Dynamics

Newton’s laws are almost universally accepted as the fundamental principles of mechan-
ics. Newton’s laws directly provide a means for studying dynamical systems. They also
provide a means for developing other principles of dynamics. These other principles are
often in forms that are more convenient than Newton’s laws for the analysis of some
classes of systems. Some of these other principles have been formulated independently
of Newton’s laws, but all of the principles are fundamentally equivalent.
The references for this chapter provide a brief survey of some of the principles of
dynamics. They include (in addition to Newton’s laws) Hamilton’s principle, Lagrange’s
equations, d’Alembert’s principle, Gibbs equations, Boltzmann–Hamel equations, Kane’s
equations, impulse–momentum, work–energy, and virtual work.
Hamilton’s principle, which is widely used in structural analyses and in approximate
analyses, states that the time integral of the difference of kinetic and potential energies of
a mechanical system is a minimum. Hamilton’s principle is thus an energy principle, which
may be expressed analytically as:

 t 2 
δ   ∫ t 1 Ldt = 0 (8.2.1)


where L, called the Lagrangian, is the difference in the kinetic and potential energies; δ
represents a variation operation, as in the calculus of variations; and t and t are any two
1 2
times during the motion of the system with t > t .
2 1
From Hamilton’s principle many dynamicists have developed Lagrange’s equations, a
very popular procedure for obtaining equations of motion for relatively simple systems.
Lagrange’s equations may be stated in the form:

d  ∂ K  ∂ K


r
dt ∂   − q ∂ r = F r = ,1 K n , (8.2.2)
q ˙
r
where K is the kinetic energy; q (r = 1,…, n) are geometric variables, called generalized
r
coordinates, which define the configuration of the system; n is the number of degrees