Velocity and Acceleration in Spherical Coordinates
Upon changing to spherical^3 coordinates (ρ, θ, φ ) the transformation equations
are
x=ρsin θcos φ, y =ρsin θsin φ, z =ρcos θ
and consequently the position vector describing the position of a moving particle is
given by
r =ρsin θcos φˆe 1 +ρsin θsin φeˆ 2 +ρcos θˆe 3 (9 .33)
Using the unit orthogonal vector ˆeρ,ˆeθ,ˆeφ in spherical coordinates obtained from
the equations (7.102) and having the representation
ˆeρ=∂r
∂ρ = sin θcos φ
ˆe 1 + sin θsin φˆe 2 + cos θˆe 3
ˆeθ=^1
ρ
∂r
∂θ
= cos θcos φeˆ 1 + cos θsin φˆe 2 −sin θˆe 3
ˆeφ=^1
ρsin θ=−sin φ
ˆe 1 + cos φeˆ 2
(9 .34)
the position vector r can be expressed in spherical coordinates by the equation
r =ρeˆρ (9 .35)
In order to obtain the first and second derivatives of equation (9.35) with respect
to time tit is necessary that one first differentiate the equations (9.34) with respect
to time t. As an exercise show that the derivatives of the equations (9.34) can be
represented
dˆeρ
dt
=∂ˆeρ
∂θ
dθ
dt
+∂ˆeρ
∂φ
dφ
dt
=dθ
dt
ˆeθ+ sin θdφ
dt
ˆeφ
dˆeθ
dt =
∂ˆeθ
∂θ
dθ
dt +
∂ˆeθ
∂φ
dφ
dt =−
dθ
dt
ˆeρ+ cos θdφ
dt
ˆeφ
dˆeφ
dt =
∂ˆeφ
∂θ
dθ
dt +
∂ˆeφ
∂φ
dφ
dt =−sin θ
dφ
dt
eˆρ−cos θdφ
dt
eˆθ
(9 .36)
One can then differentiate equation (9.35) and show the velocity in spherical coor-
dinates has the form
(^3) Note (ρ, θ, φ )gives a right-handed coordinate system, whereas the ordering (ρ, φ, θ )gives a left-handed
coordinate system. Be aware that European textbooks, many times use left-handed coordinate systems.