I N V A R I A N C E P R I N C I P L E S A N D C O N S E R V A T I O N L A W S 95
Let a Newtonian particle of mass m move in the plane about a fixed origin, O,
under the influence of a force F. The equations of motion, in the x-and y-directions, are
md
2
x/dt
2
= F
x
and md
2
y/dt
2
= F
y
. (5.5 a,b)
If the force can be represented by a potential V(x, y) then we can write
md
2
x/dt
2
= –∂V/∂x and md
2
y/dt
2
= –∂V/∂y . (5.6 a,b)
The total differential of the potential is
dV = (∂V/∂x)dx + (∂V/∂y)dy.
Let a transformation from Cartesian to polar coordinates be made using the standard linear
equations
x = rcosφ and y = rsinφ .
The partial derivatives are
∂x/∂φ = –rsinφ = –y, ∂x/∂r = cosφ, ∂y/∂φ = rcosφ= x, and ∂y/∂r = sinφ .
We therefore have
∂V/∂φ = (∂V/∂x)(∂x/∂φ) + (∂V/∂y)(∂y/∂φ) (5.7)
= (∂V/∂x)(–y) + (∂V/∂y)(x)
= yF
x
+ x(–F
y
)
= m(ya
x
– xa
y
) (a
x
and a
y
are the components of acceleration)
= m(d/dt)(yv
x
– xv
y
) (v
x
and v
y
are the components of velocity).
If the potential is independent of the angle φ then
∂V/∂φ = 0, (5.8)
in which case