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Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 243

of freedom of the system; and the F (r = 1,…, n) are generalized forces exerted on the
r
system.
Another procedure, similar to Lagrange’s equations, is Gibbs equations, which state that:

∂G
= F (r = ,1 K ) n , (8.2.3)
∂q ˙˙ r
r
where G, called the Gibbs function, is a kinetic energy of acceleration defined as:

N
G = 1 ∑ m ( a P i ) 2 (8.2.4)
R
2 i
i=1
R
where P is a typical particle of the mechanical system, m is the mass of P , a P i is the
i
i
i
acceleration of P in an inertial reference frame R, and N is the number of particles of the
i
system. An inertial reference frame is defined as a reference frame in which Newton’s
laws are valid.
A principle which we will examine and use in the remaining sections of this chapter,
called d’Alembert’s principle, is closely associated with Newton’s laws. d’Alembert’s prin-
ciple introduces the concept of an inertial force, defined for a particle as:
D
F =− m R a P i (8.2.5)
*
i i
Then, d’Alembert’s principle states that the set of all applied and inertia forces on a
mechanical system is a zero force system (see Section 6.4).
A relatively recent (1961) principle of dynamics, known as Kane’s equations, states that
the sum of the generalized applied and inertia forces on a mechanical system is zero for
each generalized coordinate. That is,

F + F = r = ,K n , (8.2.6)
*
1
0
r r
Kane’s equations combine the computational advantages of d’Alembert’s principle and
Lagrange’s equations for a wide variety of mechanical systems. For this reason, Kane’s
equations were initially called Lagrange’s form of d’Alembert’s principle.
Finally, still other principles of dynamics include impulse–momentum, work–energy,
virtual work, Boltzmann–Hamel equations, and Jourdain’s principle. We will consider the
impulse–momentum and work–energy principles in the next two chapters. The principles
of virtual work and Jourdain’s principle are similar to Kane’s equations, and the Boltz-
mann–Hamel equations are similar to Lagrange’s equations and Gibbs equations.

8.3 d’Alembert’s Principle
Newton’s second law, which is probably the best known of all dynamics principles, may
be stated in analytical form as follows: Given a particle P with mass m and a force F