What is the relation between the energy and the momentum? (Please excuse my English)
One day, I realized that I had not clearly learned about the relation between the energy and the momentum, though I am an educated engineer.
I just thought the energy is a kind of scalar transform of the momentum, in Newtonian mechanics. Also I thought the energy conservation can be derived from the momentum conservation(in Newtonian mechanics).
However, I heard that situations are more complicated in Relativistic mechanics. The energy conservation is related to time-axis symmetry. And the energy is closely related to the time-axis component of the 4-dimensional momentum.
I am confused. What is the exact relation between the energy and the momentum?

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*Added after reading comments : Yes, I also think that my understandings on the energy and the momentum are not good. That's why I posted this question, for better understandings. I just want more simple relation between the energy and the momentum which anyone can easily understand.

 A: In a non relativistic context: $E = \frac{1}{2}mv^2 = \frac{m^2v^2}{2m} = \frac{p^2}{2m}$
In a relativistic context: $E = \sqrt{m^2 + p^2}$ (setting c = 1)
A: So if you have ever seen a Newton’s Cradle toy, there is a mysterious effect where you pick up 3/5 of the balls on say the “left” side, and after they collide with the two stationary balls on the “right,” 2 balls on the left remain absolutely stationary while 3 balls fly off together on the right. How does the middle ball know to keep moving at constant speed while the first two balls know to immediately stop? Or if you only pick up 2/5 of the balls, the central ball now remains still even though it receives all manner of impact forces.
Conservation of momentum is not enough to reach this conclusion because one option which conservation of momentum allows, for instance, is 4/5 balls remaining stationary while the last ball on the right flies off with speed $2v$ or $3v$ or so.
Conservation of energy is the remaining constraint which comes to help us out: those options would clearly have $4K$ or $9K$ of the kinetic energy $K$ of any single ball, and only these eerie patterns where the exact same balls leave as entered, are the ones allowed when simultaneously conserving both. They are both conserved because the balls are big metal spheres which mostly return back to their original state right after impact. If you try to do a Newton's cradle with eggs and they shatter, probably you do not see such a nice behavior. If you try to do one with loaded mousetraps which "spring" as they collide, the same is probably true, you probably get more energy in the outside swing than is coming in.
But what is energy though?
Energy gives us a different perspective on classical Newtonian mechanics. This starts from taking Newton’s laws for a particle with mass $m$ subject to a bunch of forces $\mathbf F_i$ as
$$ m \frac{\mathrm d\mathbf v}{\mathrm dt} = \sum_{i=1}^N \mathbf F_{i}.$$
If we define the power exerted by the force as $P_i = \mathbf F_i \cdot \mathbf v$ then we get the interesting result that $$\sum_{i=1}^N P_i = m \frac{\mathrm d\mathbf v}{\mathrm dt} \cdot \mathbf v = \frac{\mathrm d\phantom t}{\mathrm dt}\left(\frac12 m \mathbf v\cdot\mathbf v\right),$$so that these powers exerted by forces determine the rate of change of the kinetic energy, which also determines the rate of change of the speed. In addition very often you can find that $\mathbf F_i = -\nabla U_i$ for some conservative force field $U_i(\mathbf r)$, and the gradient operator $\nabla.$ But, it turns out that the very definition of this gradient operator is that $U_i(\mathbf r + \delta\mathbf r) \approx U(\mathbf r) + \nabla U_i \cdot \delta \mathbf r,$ and since $\mathbf v = {\mathrm d\mathbf r}/{\mathrm dt}$, if you squint a bit you see both a $\mathrm d\mathbf r$ and a $\delta \mathbf r$ and you can realize that as long as this function $U_i$ is not changing over time then $\nabla U_i \cdot \mathbf v = \mathrm dU/\mathrm dt$ by the chain rule. If all of the forces can be described this way then we have the equation of conservation of energy, $$\frac{\mathrm d\phantom t}{\mathrm dt} (K + U) = 0.$$
So we have a really nice rock-solid principle for looking at the world but unfortunately it only gives us some of the information that we want, you cannot reconstruct the entire trajectory of a system from only the conservation of energy.
Now it happens to be the case that a really brilliant French-Italian mathematician named Joseph-Louis Lagrange came up with a rephrasing of Newton’s classical mechanics,

Of all of the trajectories that a system could take from state A to state B, the one that it does take will be one where “nearby” trajectories all have the same “action,” this being a number which we can compute from a given trajectory. (In the usual case, the action is the time-integral of a Lagrangian which is generally the difference between the kinetic and potential energies.)

You can think of the “action” or the “Lagrangian” therefore as “the laws of physics”. In this rephrasing, we use energy considerations to actually derive the equations of motion and perhaps solve for the trajectory. So the laws are entirely encapsulated in this energy function. Neat trick, that: for finally energy is not just one consideration but a full perspective on how classical mechanics works. Then another 50 years later, a brilliant Irishman named William Rowan Hamilton decided that the French did not get to have all the fun and derived a notion of “phase space” and a total-energy function, kinetic plus potential energies over that phase space, which could now give you the equations of motion directly as well.
About 75 or so years after that, a brilliant German-Jewish woman named Emmy Noether realized that Lagrange’s equations held a really important secret, which could be seen in this idea that “nearby” paths had the same value for the action on the actual trajectories. The idea was to take the idea in reverse: suppose you know that this action does not vary with some coordinate (we say that it has a symmetry). Then you might imagine trajectories which move in that symmetry direction, then look exactly like the actual trajectory, then move back: and because of the symmetry the two nearby trajectories sum up exactly the same, so the motion at the beginning must be exactly compensated by the motion at the end. And this implies that there is a rate-of-change of this Lagrangian with respect to that symmetry, which is the same at either side.
So these continuous symmetries, she said, correspond to conserved quantities, and vice versa:

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*Conservation of momentum in some direction, is the same thing as saying that the laws of physics are the same here as they are if we step one millimeter in that direction.

*Conservation of angular momentum about some axis, is the same thing as saying that the laws of physics are the same in this situation as they are if we rotate by one degree around that axis.

*Conservation of energy, is the same thing as saying that the laws of physics are the same now as one second from now.

(Notice that we already saw a hint of that, I said that $U_i$ needed to not change over time above!)
So they can't be derived from each other. Here is a situation that conserves energy but not momentum: I drop a rock. (It conserves momentum if you include the entire Earth in your calculation, but if I just assume that the ground is fixed and there is a force field $-m g \hat z$ that mathematical situation is not conserving momentum.) Here is a situation that conserves momentum but not energy: I leave that rock on the ground and then I silently remove half of the mass of the Earth so that gravity is half as strong.
Now where does relativity come in? It comes in by saying that actually they can be derived from each other. These situations that conserve momentum but not energy, conserve neither in relativity: because if you were to look at the situation from the point of view of a spaceship crashing into Earth, you would see the momentum or energy changing as well. The only true conservations in relativistic mechanics are conservations of both momentum and energy. Which is kind of a head trip!
