Tensors for Physics

80 7 Fields, Spatial Differential Operators

surface. Then one has dΦ=0, and consequently, in this case∇μΦdrμ=0. This
means: the gradient field

∇μΦ(r)is perpendicular to the equipotential surface running throughr.

It is recalled that a two-dimensional potential function can be visualized by analogy
to a map showing the height of hills and dales. The gradient points in the direction of
the steepest ascent. Hikers know, when the lines of equal height are close together, the
ascent is very steep. The interrelation between the force and the potential is defined
such that the force is determined by the negative gradient. In the landscape analogy,
the force points into the direction of the steepest descent, just as water flows.
In mechanics, there are forces which can be derived from a potential and others,
for which no potential function exists. Examples will be discussed later. When the
forceFacting on a single particle can indeed be derived from a potential, then it is
determined by

Fμ=Fμ(r)=−∇μΦ(r). (7.3)

Similarly, in electrostatics, the electric fieldE=E(r)is the negative gradient of the
electrostatic potentialφ=φ(r):

Eμ=Eμ(r)=−∇μφ(r). (7.4)

The electric field is everywhere perpendicular to the surfaces of equal electrostatic
potential. The field lines indicate these directions normal to the potential surfaces.
The electric field, at a specific point, is parallel to the tangent vector of the electric
field running through this point.

7.1.4 Newton’s Equation of Motion, One and More Particles

In Newton’s equation of motion the positionr=r(t)of a particle is a function of
the timet. For a constant massmand with the velocityv(t)=dr/dt, the equation
of motion now reads

m

dvμ

dt

=m

d^2 rμ

dt^2

=Fμ=−∇μΦ(r). (7.5)

So far, the dynamics of a single particle, subjected to an “external” force, was con-
sidered. The description of the dynamics ofN= 2 , 3 ,...particles with position
vectorsriand massesmiwherei= 1 , 2 ,...,N, is based on equations of motion
for each one of these particles:

mi

dvμi

dt

=mi

d^2 riμ

dt^2

=Fμi. (7.6)