# Why do we need the quantity momentum?

Why do we need the quantity Momentum in physics when we have the quantities like Force and Energy? Isn't it possible to substitute the usage of Momentum with equivalent of Force and Energy?

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Energy and momentum have a great deal of conceptual overlap, but they are conserved separately. There are some times when momentum is conserved but energy is not.

As for force, there are many situations where the momentum can be calculated easily, but the force is tough to find because we don't know how much time something takes. In a car crash, for instance, we might know how fast the cars were moving beforehand, but not how long they were in contact or what the force between them was. Momentum works even if we don't know those things. Some problems that would be unsolvable with force alone thus become solvable if we use momentum instead.

On the more-information-than-you-need level, all conserved quantities come from symmetries in our universe. Because the laws of physics are the same no matter what direction you look in, angular momentum is conserved. Because they're the same whether you move left or right, regular momentum is conserved. Because they're the same now as they will be later in time, energy is conserved. Energy and momentum conservation come from two separate symmetries; they're both true, so we need to keep track of both quantities.

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Isn't energy always conserved? Can you give an example? –  aortizmena May 17 '12 at 5:29
While it's true that energy can't be created or destroyed, it can end up in forms that aren't useful to us. In a car engine, for example, you use chemical energy in the fuel to give kinetic energy to parts in the engine, but some of the energy turns into heat, which isn't useful. (In fact, most of it ends up as heat; engines are not very efficient.) In introductory physics we often talk about energy "lost" to friction or deformation. –  Colin Fredericks May 17 '12 at 13:35
So it is conserved but not useful for calculations sometimes. I asked due to your 2nd statement. –  aortizmena May 18 '12 at 6:15
Energy can be 'lost' from the system you're looking at; e.g. if I'm looking at a bouncing ball, it loses some of its energy to the surface it's bouncing off (sound, heat etc.), and in some sense we don't care about what's happening to the surface, so the system we're looking at (the ball) is losing energy. –  Daniel Blay Jul 5 '12 at 23:59

The only thing you really need in classical mechanics is Newton’s Laws. Energy and momentum conservation are just principles derived from them. That is why they are not called laws. If you integrate, with respect to displacement, both sides of Newton's 2nd law in the presence of forces that can be expressed as the negative gradient of some function (i.e. integral independent of pass) you gey classical conservation of energy. Momentum conservation is derived from Newton's third Law. Canonical momentum is not needed in lagrangian formulation of mechanics. Canonical momentum enters Hamilton's equations as a Legrende transformation in phase space. To answer your question, only force and mass is needed for a complete description of a classical particle. Energy and momentum are just constants of the motion only when there are no external forces on the system and no dissipative forces.

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Colin did a good job answering this question but I can't help fill in some other things. As for the utility of force, momentum and energy:

First of all, when you are trying to solve a physics problem, you are never going to keep anything off the table - for difficult problems you need all the tools at you disposal. Rarely are do we have the luxury of selecting amongst a variety of ways to solve a problem - we do what we can with what we have, which in some cases may mean using the quantity of momentum is our only way forward.

Second of all, off hand, I honestly don't particularly care what momentum is, but there are some cases, namely when the net force on an object is zero, that momentum is conserved - and that is a very powerful statement. Without any knowledge of the details of say, the collision , the composition or structure of the objects, we can immediately say that $\vec{p}_i = \vec{p}_f$ which is a very useful statement. The case is similar for angular momentum, I could care even less what that is for a general system, but if the net torque on a system is zero and angular momentum is conserved you better believe I am using it if it helps solve me solve the problem.

Thirdly, energy is conserved, and thus particularly useful, when the net work done on a system is zero. This a completely different case than when momentum is conserved, so these are clearly not redundant concepts, although there may be times when both are aplicable.

Finally, there is simply no useful notion of 'force' in quantum systems in general. Yes you can do some calculations and sometimes handwave what a force is in a certain situation, but in most matters its of little particular use. Take the collisions at the LHC - there is no useful notion of 'force' you can ascribe to what is going on when all those particles collide. Yes, if you stand in the LHC beam pipe you will feel a force, and if you survive you can tell us about it, but from a standpoint of predicting what so-and-so particle got scattered into what angle, or what the lifetime of some particle is, force is never mentioned. As another example, take the spectrum of the hydrogen atom - we obtain this by solving for the allowed energy values of the Schrodinger equation, and again force is not useful and has nothing to say about the problem for that matter.

In conclusion, if you want to do real physics its all about utility. Use what you can to solve the problem, and momentum is a useful quantity.

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