# Can potential energy be the sum of the work done by all forces?

I often see that the total energy of a system is the sum of potential energy+ kinetic energy. Is it always like that? Could I say that the total energy of a system is the sum of the work of all force + kinetic energy? I'm always confuse on which force gives a potential or not... But maybe at the end, it's just the sum of work of each force?

• Body has energy of movement, that is kinetic energy. Potential energy is defined as a work of certain (so-called conservative) forces to a body. So total energy in mechanics is simply the potential energy plus kinetic energy. – Pygmalion Dec 15 '18 at 18:20
• @Pygmalion : why don't w consider the work of the non conservative force in the total energy ? – user623855 Dec 15 '18 at 18:21
• Because it turns into other non-mechanical forms of energy (internal energy, heat...) – Pygmalion Dec 15 '18 at 18:25

Could I say that the total energy of a system is the sum of the work of all force + kinetic energy ?

It is not useful to do this because the work done by a force is not a state function. What this means is that we cannot define a unique "work done by all forces" of a system based on the state of the system alone. You would have to know how the forces acted on the system in the past, and even then you would have to define a starting time and initial energy of the system to using this method. In other words, the total energy would depend on when we decide to start adding up all of the work that has been done on the system.

The reason we define the total energy as $$E=K+U$$ is because of how we define these energies and forces to make life easier. I will consider a single particle, but this can be generalized to more complicated systems. We know that the net work on a particle (over some time) determines its change in kinetic energy:

$$W_{net}=\Delta K$$

But we can also express the net work as work done by a sum of work done by conservative and non-conservative forces:

$$W_{net}=W_c+W_{nc}$$

But by definition of conservative forces and their associated potential energies, we know that

$$W_c=-\Delta U$$

Therefore, if we define our total energy as $$E=K+U$$, then we see that

$$\Delta E=\Delta K+\Delta U=W_{nc}$$

So we find that only non-conservative forces can change what we have defined as total energy. Furthermore, if there are no non-conservative forces acting on the particle, we see that total energy is conserved.

Going back to my first point, we see that our definition $$E=K+U$$ only depends on the state of the system (typically just the position and velocities of particles). It doesn't depend on the history of the system. Notice how work only determines changes in energy, never actual energy values.

I'm always confuse on which force gives a potential or not

Conservative forces have a potential energy associated with them. Conservative forces are forces that have zero curl, or in other words are forces where the work done by that force does not depend on the path through space the system takes. When this is the case, we can write out a potential energy such that $$\mathbf F=-\nabla U$$ Or in one dimension $$F=-\frac{\text d U}{\text d x}$$

• Being relative to starting time is not much worse than simply relative to chosen zero point of energy. And potential energy (and thus total energy) does have this freedom. – Ruslan Dec 15 '18 at 21:13
• @Ruslan I see what you mean. I'll think about it more and edit my answer tomorrow. – Aaron Stevens Dec 16 '18 at 4:11

For a given system the total energy is described as the energy contained in the movement of the components of the system and a potential enery given by the conservative forces that affect the system. Other forces normally make the system either lose or gain energy so those are not accounted for in the system directly (refer to "heat" in thermodynamics) and if you consider relativistic effects you should also account for rest-mass energy with Einstein's equation.

This question touches on a deep subject with many facets. The other answers already posted represent some of those facets.

I'll briefly mention one more: Noether's theorem. This theorem says that in a model that respects the action principle and that is symmetric under time-shifts, there is automatically a conserved quantity. We call it "energy." In Newtonian mechanics, this energy ends up being a sum of kinetic terms and potential terms. Other models might not have such a natural separation into "kinetic" and "potential" parts, even though they still have a conserved total energy defined via Noether's theorem.

In electrodynamics, the force due to a magnetic field on a moving charged particle is proportional to the particle's speed but perpendicular to its velocity. This force is not the gradient of any potential energy in the more-familiar Newtonian sense, but electrodynamics still has a conserved total energy defined via Noether's theorem. Whether or not we try to separate this into "kinetic" and "potential" terms is a matter of taste and convenience. The total energy is what matters in this context, because the total energy is what's conserved.

This definition based on Noether's theorem is just one of several related-but-different meanings of the word "energy" in physics. It's a deep subject, and the fact that the word is overloaded can make navigating the subject even more challenging.

It is in fact the work done by all the forces plus some constants. These constants are just formalisms which are of no use in dynamical problems. In fact you don't even need the potential formalisms as long as you make sure to include the work done by those forces separately. In that case the work done on the system would correspond just to the change in kinetic energy of the system. The potential concept is there so that you don't need to integrate the force with x again and again, you can just use a formula. As long as you don't want to lose the sense of causality, I recommend not to use the potential; you can still get the same results by using only kinetic energy but including work by all forces.