Equation of Motion in Polar Coordinates

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To analyze central force motion, it is convenient to express the equations of motion in polar coordinates (r, \theta), since the force depends only on radial distance and points along the radius vector.

Position and Velocity in Polar Coordinates

The position vector in polar coordinates is:

\mathbf{r} = r , \hat{\mathbf{r}}

The velocity vector is given by:

\mathbf{v} = \frac{d\mathbf{r}}{dt} = \dot{r} , \hat{\mathbf{r}} + r , \dot{\theta} , \hat{\boldsymbol{\theta}}

where:

  • \dot{r} = \frac{dr}{dt} is the radial velocity.
  • \dot{\theta} = \frac{d\theta}{dt} is the angular velocity.

Acceleration in Polar Coordinates

The acceleration vector is:

\mathbf{a} = \left( \ddot{r} - r , \dot{\theta}^2 \right) \hat{\mathbf{r}} + \left( r , \ddot{\theta} + 2 \dot{r} , \dot{\theta} \right) \hat{\boldsymbol{\theta}}

It has two components:

  • Radial acceleration:

a_r = \ddot{r} - r , \dot{\theta}^2

Transverse (angular) acceleration:

a_{\theta} = r , \ddot{\theta} + 2 \dot{r} , \dot{\theta}

Newton’s Second Law for Central Force

Since the central force has no transverse component:

F_{\theta} = 0

This yields:

m , a_{\theta} = 0

So:

r , \ddot{\theta} + 2 \dot{r} , \dot{\theta} = 0

This equation represents conservation of angular momentum. It can be integrated to give:

r^2 , \dot{\theta} = h

where h is a constant (specific angular momentum per unit mass).

Radial Equation of Motion

The radial component of Newton’s second law is:

m , a_r = F(r)

Substituting:

m \left( \ddot{r} - r , \dot{\theta}^2 \right) = F(r)

Using \dot{\theta} = \frac{h}{r^2}, we get:

\ddot{r} - \frac{h^2}{r^3} = \frac{F(r)}{m}

This is the radial equation of motion for central force motion.

Summary of Equations

  • Conservation of angular momentum:

r^2 , \dot{\theta} = h

Radial equation of motion:

\ddot{r} - \frac{h^2}{r^3} = \frac{F(r)}{m}

These equations fully describe the dynamics of a particle under a central force in a plane, enabling analysis of orbital trajectories and properties.

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