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I would like to clarify my question. I have numbered them to be independent questions

  1. For any conservative fields, $\vec{F} = -\nabla U$. Which means the restoring force is opposite to the increasing values of the potential energy. If it so happens that $F = \nabla U$, does that mean the restoring force (not even called that anymore I guess) would be in the same direction as the increasing values of energy? Is this a violation to the first law in thermodynamics.

  2. For attracting bodies, the closer the objects come together, the potential energy drops as $r \to 0^-$ and $U \to -\infty$, but for repelling bodies we get $r \to 0^-$ as $U \to \infty$ which means the potential energy actually increase. Am I right?

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Welcome to Physics.SE! Could you please state your question a little more clearly? – dmckee Jan 8 '12 at 0:09
Hmm you are right. I can't put it all in this commentbox. But one question is, for repelling bodies, will their potential energy increase or decrease as they come together? How will this affect the electric potential? – lem Jan 8 '12 at 0:13
You should edit that into the text of the question. Given how much text you have you might want to highlight it a bit – dmckee Jan 8 '12 at 0:45
I totally don't understand what you are asking. – Siyuan Ren Jan 8 '12 at 0:54
You seem to misinterpret what the equations say and how their solutions are used. The solutions at a given time and space are a snapshot. They do not contain a change in energy in any form. Taking the difference as you are doing and finding a change in energy between the two solutions, tells one that if energy conservation is not to be violated something else is taking it up or giving it in. In your example the kinetic energy changes, and becomes potential energy(repulsive), or the potential energy turns into kinetic energy (attractive), always conserving energy. – anna v Jan 8 '12 at 6:13

You are confused because the potential energy for repelling bodies is positive, and for attractive bodies is negative, but in both cases $F = -\nabla U $. The force always points towards decreasing potential energy. There is no force which is in the direction of increasing potential energy, because of the reason you state--- such a force would push things to have more kinetic energy at the same time as they get more potential energy, violating conservation of energy.

Flipping the sign of U means that you flip the sign of the force, but the force is still in the direction of maximally decreasing potential.

As for question 2, you are right.

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