Conservation of Energy and Momentum in Classical Physics

Conservation of Energy and Momentum in Classical Physics

Conservation of energy and conservation of momentum are often talked about as two separate principles in classical physics.  After Special Relativity it has been understood that energy-momentum is a single quantity, a four vector that transforms under the Lorentz transform.  However even in classical mechanics energy and momentum are related.

Consider two particles with masses m1 and m2 that collide with initial velocities v_1_i and v_2_i in inertial frame F1.  Let’s assume that in this frame we have conservation of energy.  The initial total energy is

After these particles collide, we must have the same total energy:

Where the velocities v_1_f and v_2_f are the final velocities after the collision.  We have one equation with two unknowns; there is no unique solution with just conservation of energy.

Now let’s require a relativity principle.  First consider Galilean relativity: that the final energy must equal the initial energy in any inertial reference frame.  Consider an inertial frame  that has velocity  wrt .  This now gives a second equation:

Expanding the products gives

This can be separated into three parts:

Now the first four terms are seen to be the conservation of energy equation and is identically zero.  The last four terms are the difference in center of mass energy before and after the collision, which is also zero.  This leaves

Which is the conservation of momentum principle.  Thus we see that by assuming conservation of energy and the Galilean relativity principle, conservation of momentum is implied.  In fact, these three principles form a triad.  If you assume any two, the third must also be true.