Advanced Classical Mechanics/Energy and Angular Momentum

Let's start with some basic definitions. First m is the inertial mass of a particle -- we will consider only particles with constant masses (although the physics of rockets provides an important counter-example). A particle for us is something that can be described completely by its position and mass (it doesn't have any important internal degrees of freedom).

Momentum and ForceEdit

We can define the velocity and momentum of the particle,

  and  .

Because the acceleration is the rate of change of velocity we have

 

so if the force vanishes, then the momentum is constant in time or conserved.

Angular Momentum and TorqueEdit

Let's also define the angular momentum and the torque,

  and  

Let's calculate the rate of change of the angular momentum

 

so if the torque vanishes, the angular momentum is conserved. Specifically if the force is always directed along the position vector,  , the torque vanishes and the angular momentum is conserved. This is called a central force.

Work and EnergyEdit

We can define the work done on a particle while moving it from position #1 to position #2 to be

 

Again let's assume that the mass of the particle is constant so

 

where we have defined the kinetic energy,

 .

If the integral   depends only on the locations of the points 1 and 2 and not the path, then the force or system is conservative. In particular,

  for such a system.

This means that the force is the gradient of some other function

 

call the potential energy  . In this case we find that

 

so

 

This is called the conservation of mechanical energy.