How can I prove that the four-momentum of relativity is conserved like in classical mechanics? The four-momentum in special relativity is defined as:
$$p = \bigg(\frac{E}{c},\ p_x,\ p_y,\ p_z\bigg)$$
where
$$p_x = \gamma (mv_x)$$
$$p_y = \gamma (mv_y)$$
$$p_z = \gamma (mv_z)$$
When solving problems, it is assumed that the individual components of this four-vector at any time will have a constant value. In other words, we assume that this quantity is conserved. How can we prove this fact?
 A: Consider the action of a free relativistic particle of mass $m$, given by,
$$S = -m\int dt \, \sqrt{1-\dot x^2}$$
and we can find the Euler-Lagrange equations imply,
$$\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot x} = \frac{\partial}{\partial t} \frac{m\dot x}{\sqrt{1-\dot x^2}} = 0$$
since $\partial L/\partial x = 0$. Note that we can identify $\partial L/\partial \dot x = p = \gamma m\dot x$. Thus, the Euler-Lagrange equations are the statement that the relativistic momenta are conserved. The Hamiltonian is given by a Legendre transform,
$$H = p\dot x - L = \sqrt{m^2 + p^2}$$
and one can see that,
$$\frac{\partial H}{\partial t} = \frac{p \dot p}{\sqrt{m^2+p^2}} = 0$$
since $\dot p =0$ by the Euler-Lagrange equations implying conservation of momentum. We thus have that the energy and relativistic momenta are conserved and so $p^\mu$ itself is conserved.

Both of these conservation laws are a consequence of the fact the system is invariant under translations in time, and the action does not depend on the position of the particle, since only its derivative $\dot x$ appears in the action. By Noether's theorem, this implies energy and momentum conservation. 
A: If a Lagrangian has the symmetry to be translationally invariant, by Noether's theorem, you can show the corresponding conserved quantities are energy and momentum. 
A: There is no a priory way of showing that 4-momentum as defined is conserved.  It is however a reasonable guess and, more importantly, it is experimentally verified that this definition leads to conservation of 4-momentum. 
I can't for the life of me immediately think of another reasonable definition - it's a 4-vector, it reduces to the usual conservation in the limit of small velocities, etc - but if this definition had failed the experimental test someone would have come up with another definition compatible with experiment.
