Elastic Force

It may be helpful to see gravity, friction, and work-energy before this.

An elastic force is any force under which energy is conserved.

The work done on a particle under an elastic force to make it travel from one point to another does not depend on the path the particle takes.

Gravity is an elastic force. (local gravity in this case) For example, compare a particle of mass $m$ traveling from A to B to C with a particle of mass $m$ traveling from A directly to C:

Let the height from A to C be $H_1$ and let the height from A to B be $H_2$. The work done on the first particle is: $$W=W_{AtoB}+W_{BtoC}=F_1d_1+F_2d_2=mgH_1+mg(H_2-H_1)=mgH_2$$ The work done on the second particle is: $$W=Fd=mgH_2$$ This is in no way a proof, but we see that the different paths lead to the same total work.

On the other hand, we can also have a force such as friction. Compare a particle of mass $m$ traveling from A to B to C with a particle of mass $m$ traveling from A directly to C:

Let the distance from A to C be $d_1$ and let the height from A to B be $d_2$. The work done on the first particle is: $$W=W_{AtoB}+W_{BtoC}=F_1d_1+F_2d_2=\mu{}mgd_1+\mu{}mg(d_1-d_2)=2\mu{}mgd_1-\mu{}mgd_2$$ (note that $d_1>d_2$) The work done on the second particle is: $$W=Fd=\mu{}mgd_2$$ We see that the different paths lead to the different total work.

See here for conservation of energy