University Physics Notes: Classical Mechanics – Proof of The Law of Conservation of Momentum
It is not obvious why momentum should be conserved. Momentum is a very abstract concept, and the principle of conservation of momentum may be seen as a consequence of other, fundamental Laws of physics:
The principle of Relativity
The Principle of Conservation of Energy
In the inertial frame O' above the particle is moving with
velocity
in
the y' direction, and it's kinetic energy is
and
if the particle moves in a potential field V due to particles at
positions relative to
of
it
has potential energy
In
the inertial frame O', the velocity of the particle is
and
it's kinetic energy is
and
the Galilean transformation
gives
so
that
The sum of all the energies in each inertial frame by the Law of Conservation of energy.
Hence the difference of the two energies is constant.
is
constant and so is
therefore
is
a constant vector.