Derivatives and Forces
john.erich.ebner@gmail.com
http://blackholeformulas.com/
15 June 2018

Parametric derivatives.
Plane polar coordinates.
r = (x2 + y 2)1/2
theta = Θ or θ = tan-1 y/x
x is a function. x' is a derivative.
vel = velocity or angular velocity.
acc = acceleration or angular acceleration.

x = r cos θ
y = r sin θ
r is a variable not a constant.

x' =
q' s +
q s'

x' = (r)' cos θ + r (cos θ)'
= r' cos θ - r sin θ vel

y' = (r)' sin θ + r (sin θ)'
= r' sin θ + r cos θ vel

x'' =
q' s +
q s' +
t' u v +
t u' v +
t u v'

x'' =
(r') cos θ + r' (cos θ) =
1 = r'' cos θ - r' sin θ vel +
- (r) sin θ vel - r (sin θ) vel - r sin θ (vel) =
2 = - r' sin θ vel - r cos θ vel 2 - r sin θ acc

x'' = 1 + 2
= r'' cos θ - r' sin θ vel + r' sin θ vel - r cos θ vel 2 - r sin θ acc
= r'' cos θ - r cos θ vel 2 - r' sin θ vel - r' sin θ vel - r sin θ acc
= [r'' - r vel 2 ] cos θ - [ 2 r' vel + r acc ] sin θ

[r'' - r vel 2 ] = acceleraions in the r direction
r'' and r vel 2 = the centripetal acceleration
[ r acc + 2 r' vel ] = acceleraions in the θ direction
acc is the angular acceleration and 2 r' vel = coriolis acceleration

y'' =
(r') sin θ + r' (sin θ) = r'' sin θ + r' cos θ vel +
(r) cos θ vel + r (cos θ) vel+ r cos θ (vel) = r' cos θ vel - r sin θ vel vel + r cos θ vel
= r'' sin θ + r' cos θ vel + r' cos θ vel - r sin θ vel 2 + r cos θ vel
= r'' sin θ - r sin θ vel 2 + 2 r' cos θ vel + r cos θ vel
= [r'' - r vel 2] sin θ + [2 r' vel + r vel ] cos θ

Ref: Classical Mechanics, Corben and Stehle, Dover 1977, page 13

Parametric derivatives.
Spherical polar coordinates.
r = (x2 + y 2 + z 2)1/2
theta = Θ or θ = cot-1 z/(x2 + y 2)1/2
phi = Φ or φ = tan-1 y/x
x is a function. x' is a derivative.
vel = velocity or angular velocity.
acc = acceleration or angular acceleration.

x = r sin theta cos phi
y = r sin theta sin phi
z = r cos theta

x' = r sin theta cos phi
= (r) sin theta cos phi
1 = r' sin theta cos phi +
x' = r (sin theta) cos phi
2 = r cos theta theta' cos phi +
x' = r sin theta (cos phi)
3 = -1 r sin theta sin phi phi'

x' = 1 + 2+ 3
x' = r' sin theta cos phi + r cos theta theta' cos phi -r sin theta sin phi phi'

y = r sin theta sin phi
y' = (r) sin theta sin phi
1 = r' sin theta sin phi +
y' = r (sin theta) sin phi
2 = r cos theta theta' sin phi +
y' = r sin theta (sin phi)
3 = r sin theta cos phi phi'

y' = 1 + 2 + 3
y' = r' sin theta sin phi +r cos theta theta' sin phi +r sin theta cos phi phi'

z = r cos theta
z' = (r) cos theta
1= r' cos theta
z' = r (cos theta)
2 = r -1 sin theta theta'

z' = 1 + 2
z' = r' cos theta - r sin theta theta'