This means that Newton’s second law can be writtenFmd
dt(^2) x
2 (9.3)
It is not uncommon to ignore the vector character of force and position and
express equation 9.3 as
Fmd
dt
(^2) x
2
Note two things about Newton’s second law. First, it is a second-order ordi-
nary differential equation.†This means that in order to understand the motion
of any object in general, we must be willing and able to solve a second-order
differential equation. Second, since position is also a vector, when we consider
changes in position or velocity or acceleration we are not only concerned about
changes in the magnitude of these values but changes in their direction.
A change in direction constitutes an acceleration since the velocity, a vector
quantity, is changing its direction. This idea has serious consequences in the
consideration of atomic structure, as we will see later.
Though they took time to be accepted by contemporary scientists, Newton’s
three laws of motion dramatically simplified the understanding of objects in
motion. Once these statements were accepted, simple motion could be studied
in terms of these three laws. Also, the behavior of objects as they moved could
be predicted, and other properties such as momentum and energies could be
studied. When forces such as gravity and friction were better understood, it
came to be realized that Newton’s laws of motion properly explained the mo-
tion ofallbodies. From the seventeenth through the nineteenth centuries, the
vast applicability of Newton’s laws of motion to the study of matter convinced
scientists that all motion of all physical bodies could be modeled on those
three laws.
There is always more than one way to model the behavior of an object. It is
just that some ways are easier to understand or apply than others. Thus,
Newton’s laws are not the only way of expressing the motion of bodies. Lagrange
and Hamilton each found different ways of modeling the motion of bodies. In
both cases the mathematics of expressing the motion are different, but they are
mathematically equivalent to Newton’s laws.
Joseph Louis Lagrange, a French-Italian mathematician and astronomer
(Figure 9.2), lived a hundred years after Newton. By this time the genius of
Newton’s contributions had been recognized. However, Lagrange was able to
make his own contribution by rewriting Newton’s second law in a different but
equivalent way.
If the kinetic energy of a particle of mass mis due solely to the velocity
of the particle (a very good assumption at that time), then the kinetic en-
ergy Kis
Km
2
( ̇x^2 y ̇^2 z ̇^2 ) (9.4)
where ̇xdx/dt, and so on. (It is a standard notation to use a dot over a vari-
able to indicate a derivative with respect to time. Two dots indicates a second
9.2 Laws of Motion 243
†Recall that an ordinary differential equation (ODE) has only ordinary, but not partial,
differentials, and that the order of an ODE is the highest order of the differentials in the
equation. For equation 9.3, the second derivative indicates a second-order ODE.
Figure 9.2 Joseph Louis Lagrange (1736–
1813). Lagrange reformulated Newton’s laws in a
different but equivalent way. Lagrange was also
an astronomer of some repute. In 1795, he and
several other prominent French scientists devised
the metric system.
© CORBIS-Bettmann