Let r be the radius vector of a particle from some given
origin and v its vector velocity:
The linear momentum p of the particle is defined as
the product of the particle mass and its velocity:
p = mv Eq. 2
In consequence of interactions with external objects and
fields, the particle may experience forces of various types, e.g.,
gravitational or electrodynamics; the vector sum of these forces exerted on the
particle is the total force F. The mechanics of the particle is
contained in Newton's second law of motion, which states that there exist
frames of reference in which the motion of the particle is described by the
differential equation
In most instances the mass of the particle is constant and
Eq. (4) reduces to
Where a is the vector acceleration of the particle
defined by
The equation of motion is thus a differential equation of
second order, assuming F does not depend on higher-order derivatives. A
reference frame in which Eq. (3) is valid is called an inertial or Galilean
system. Even within classical mechanics the notion of an inertial system is
some-thing of an idealization. In practice, however, it is usually feasible to
set up a coordinate system that comes as close to the desired properties as may
be required. For many purposes, a reference frame fixed in Earth (“the laboratory
stem") is a sufficient approximation to an inertial system. While for some
astronomical purposes it may be necessary to construct an inertial system by
reference to distant galaxies. Many of the important conclusions of mechanics
can be expressed in the form of conservation theorems, which indicate under
what conditions various mechanical quantities are constant in time. Equation (3)
directly furnishes the first of these, “the Conservation Theorem for the Linear
Momentum of a Particle: If the total force. F, is zero, then p =
0 and the linear momentum p, is conserved.
The angular momentum of the particle about point 0, denoted
by L, is defined as
L = r x p Eq. 7
Where r is the radius vector from 0 to the particle.
Notice that the order of the factors is important. We now define the moment of
force or torque about 0 as
N = r x F Eq. 8
The equation analogous to (3) for N is obtained by
forming the cross product of r with Eq. (4):
Equation (9) can be
written in a different form by using the vector identity:
Where the first term
on the right obviously vanishes. In consequence of this identity. Eq. (9) takes
the form
Note that both N and L depend on the point 0, about
which the moments are taken.
“Conservation Theorem for the Angular Momentum of a
Particle: If the total torque, N, is zero then L = 0, and the
angular momentum L is conserved. “
Next consider the work done by the external force F
upon the particle in going from point 1 to point 2. By definition, this work is
For constant mass (as will be assumed from now on unless
otherwise specified), the integral in Eq. (12) reduces to
The scalar quantity mv2/2 is called the kinetic energy of the particle and is denoted by T, so that the work done is equal to the change in the kinetic energy:
W12 = T2 - T1 Eq.14
If the force field is such that the work WI, is the same for any physically possible path between points 1 and 2, then the force (and the system) is said to be conservative. An alternative description of a conservative system is obtained by imagining the particle being taken from point 1 to point 2 by one possible paths and then being returned to point 1 by another path. The independence of W12 on the particular path implies that the work done around such a closed circuit is zero, i.e.:
Physically it is clear that a system cannot be conservative if friction or other dissipation forces are present, because F.ds due to friction is always positive and the integral cannot vanish. By a well-known theorem of vector analysis, a necessary and sufficient condition that the work. Wit be independent of the physical path taken by the particle is that F be the gradient of some scalar function of position:
Where V is called the potential energy. The existence of V can be inferred intuitively by a simple argument. If W12 is independent of the path of integration between the end points 1 and 2, it should be possible to express W12 as the change in a quantity that depends only upon the positions of the end points. This quantity may be designated by -V, so that for a differential path length we have the relation
F • ds = —dV
or
Which is equivalent to Eq. (16)? Note that in Eq. (16) we can add to V any quantity constant in space, without affecting the results. Hence the zero level of V is arbitrary.
For a conservative system, the work done by the forces is
W12 =V1 — V2 Eq. 17
Combining Eq. (17) with Eq. (14), we have the result
T1 + V1 = T2 + V Eq. 18
Which states in symbols the “Energy Conservation Theorem for a Particle: If the forces acting on a particle are conservative, then the total energy of the particle, T + V, is conserved.”
The force applied to a particle may in some circumstances be given by the gradient of a scalar function that depends explicitly on both the position of the particle and the time. However, the work done on the particle when it travels a distance ds,
is then no longer the total change in —V during the
displacement, since V also changes explicitly with time as the particle
moves. Hence, the work done as the particle goes from point 1 to point 2 is no
longer the difference in the function V between those points. While a total
energy T + V may still be defined, it is not conserved during the course
of the particle's motion.