Advanced Classical Mechanics/Many-Particle Systems

Here we will look a group of particles with forces that exert forces among themselves and have external forces exerted upon them as well. If we focus on particle , we have

where the first term is the force that particle exerts on particle summed over the other particles. The second term denotes the external force on particle .

Weak law of action and reaction -- total momentumEdit

Let's assume the weak law of action and reaction which states that   and sum the change in the momentum of all the particles

 

The second term vanishes because each force between each pair of particles cancels out. If we define the position of the center of mass of the system as

 

we have

 

The total momentum is

 

so if   then   is conserved.

Strong law of action and reaction -- total angular momentumEdit

The total angular momentum of the system is given by

 

and its rate of change is

 

where the first term vanishes because the momentum and velocity point in the same direction.

Substituting in the forces yields

 

If we look at the final term more closely we have

 

so if we assume that the force between the particles points in the same direction as the displacement between the particles (strong law of action and reaction), the forces between the particles do not affect the total angular momentum of the system. This type of force is called a central force. In this case we have   so

 

A central force satisfies the strong law of action and reaction (not necessarily the weak law). Gravity and electric forces are central and also satisfy the weak law of action and reaction but magnetic forces generally satisfy neither.

Let's take a closer look at the total angular momentum of the system of particles by defining the displacement of a particle relative to the center of mass.

 

The total angular momentum is given by

 

The two middle terms vanish because

 

from the definition of the center of mass. We are left with

 

That is the angular momentum of a system is the angular momentum of the center of mass plus the angular momentum about the center of mass.

Uniform Gravitational FieldEdit

A familar possibility for the external forces and torques is a uniform gravitational field. In this case we have

 

Let's calculate the change in the velocity of the center of mass

 

and the change in the angular momentum

 

so

 

In an uniform gravitational field, the internal angular momentum of a body is conserved. Think about footballs and frisbees!

Work and Kinetic EnergyEdit

Let's examine the total work performed on the system of particles. We have

 

We also have

 

so

 

We can also look at the energy using the center of mass to get

 

The middle term vanishes for the reasons outlined earlier.

Potential EnergyEdit

First let's look at the external forces

 

where   if the external forces are conservative. N.B.: in this section,   denotes potential energy not the velocity of the center of mass.

To look at the interparticle forces, let's assume that they too are conservative and that the potential only depends on the distance between the particles so

 

Such a potential has two nice properties. First,

 

so it satisfies the weak law of action-reaction. Second we have

 

so it is central and satifies the strong law of action-reaction. If we use this particular type of force we have

 

Let's define  . We have

 

and

 

If we substitute these results into the expression for the work we have

 

so we can definte the total potential energy to be the sum of the potential energies of each particle due to the external force plus the sum of the potential energies of each pair of particles:

 

The Virial TheoremEdit

Let's look at particular and common form of central force. In particular, let's assume that

 

and that there is no external potential. This means than

 

We can define

 

as a measure of the size of the system. Let's calculate the second derivative of   with respect to time

 

Again let's group the particles in the final summation in pairs

 

If we have a bunch of particles in equilibrium so they are not expanded or contracting on average than the left-hand side of the equation above vanishes. The most familar example of this is the force of gravity with  , so for a self-gravitating system in equilibrium we have  . This relationship is used to weigh distant galaxies and clusters of galaxies.

If   the potential energy of the system vanishes as the particles get further apart, so the total energy of the system   is the negative of the amount of energy needed to break up the system and leave the particles stationary. Let's calculate the total energy of a system in equilibrium in terms of its kinetic energy. We have

 

We find that only if   is the energy of the system negative. For   the total energy of the equilibrium system is zero, so the equilibrium is marginally stable; even a tiny bit of additional energy is enough to break it apart. For   the equilibrium is unstable. The grouping of particles has more energy than if they separate.

Thankfully the force of gravity has  . If   were less than -2, there would be no self-gravitating equilibrium systems such as stars, galaxies etc. In fact theories of gravity in higher dimensions find that   where   is the number of spatial dimensions. Apparently   is kind of special.