Tensors for Physics

7.1 Scalar Fields, Gradient 81

Clearly,viis the velocity of particle “i”. When the force can be derived from a
potential,thismany-particlepotentialΦisafunctionofthecoordinatesofallparticles
involved, viz.:Φ=Φ(r^1 ,r^2 ,...,rN).TheforceFiacting on particle “i”isgiven
by the negative partial derivative of the potential with respect to the vectorri:

Fμi=−

∂

∂rμi

Φ=−∇iμΦ. (7.7)

An important special case are two interacting particles,N=2. When their dynam-
ics can be treated as if they were isolated from the rest of the world, the rele-
vant interaction potential depends on their positions only via the relative vector
r^12 =r^1 −r^2. Then one has∂Φ/∂rμ^1 =∂Φ/∂rμ^12 and∂Φ/∂r^2 μ=−∂Φ/∂r^12 μ.This

impliesFμ^1 =−Fμ^2 , which corresponds to Newton’sactio equal reactio. As a con-
sequence, the motion of the center of mass of the two particles is “force free”, and
the total linear momentumP=m 1 v^1 +m 2 v^2 is constant. The interesting motion is
that one described by the relative vectorr^12 and the relative velocityv^12 =v^1 −v^2.
The governing equation of this motion is

m 12

dvμ^12

dt

=m 12

d^2 rμ^12

dt^2

=Fμ^1 =−

∂Φ(r^12 )

∂rμ^12

, (7.8)

with the reduced massm 12 =m 1 m 2 /(m 1 +m 2 ). Thus the two-particle dynamics is
reduced to the force-free motion of its center of mass and an effective one-particle
dynamicsoftherelativemotion.Withthereplacementsm 12 →m,r^12 →r,F^1 →F,
Φ(r^12 )→Φ(r)and(∂/∂rμ^12 )→∇μthe equation of motion (7.8) is mathematically
equivalent to (7.5). In this sense, some of the special potential and force functions to
be mentioned below pertain to an effective one-particle problem rather than to a true
one particle dynamics.

7.1.5 Special Force Fields

The application of the nabla operator to the vectorryields

∇μrν=δμν. (7.9)

The contraction withμ=νimplies

∇μrμ= 3 , (7.10)

in 3D. The fact that the scalar product of the nabla operator with the vectorryields
a number, which is certainly a scalar, proves that the nabla operator is also a vector.
The relation (7.9) is essential for the calculation of the force from a given potential