# Are there any non-dissipative non-conservative forces?

Although the question stands for itself, I would like to know that if the answer has to be no then does any particular law forbids the existence of such forces; and if there are such forces then what are these and upon travelling under such forces in a closed loop where does the energy come from (as they are non conservative and non dissipative)?

Addendum: I would also highly appreciate it if someone could give an example of force fields which could be set up and left undisturbed by movement of particles under their influence. For example a charged non conducting fixed sphere provides an approximately fixed static field under which if particles move the cause is not disturbed. Since we know about induced electric field, but motion of charged particles under induced electric field, disturb its cause and then we have to supply more energy to maintain the field at original state.

I would hasten to add that although this would be highly appreciated, its not a necessity for the original question.

• The only case where I think this could be true is when there is an electromagnetic field varying with time. In General Relativy energy is not conserved (it's ill-defined, in fact). But in that case gravity is not a force anymore. – jinawee Dec 27 '13 at 17:57
• Time-varying E&M field would be a good example indeed - say the electric field generating the loop voltage in the secondary winding of a transformer. The electric field can be made constant in time as long as constant dB/dt can be maintained. – Maxim Umansky Dec 27 '13 at 18:03
• So lets just say for the sake of argument, if there was any non conservative non dissipative force then work done over a loop would be positive and it will violate energy conservation, so should such forces be classified as surreal or can we safely say that if we do find such a force we can say that energy conservation has an exception for this particular type of force ? – Rijul Gupta Dec 27 '13 at 18:12
• No, no. Energy is always conserved (except the marginal case in GR). Work around a closed loop $\neq 0$ doesn't imply that energy is not conserved. – jinawee Dec 27 '13 at 18:29
• Then where would the energy be coming from, clearly if we set up a constant field in which the forces are non dissipative and non conservative then there would be positive work in one loop, also since the field is constant, and this motion is not changing the field, the motion can go on and on, accumulating energy. How then would enegy be conserved ? – Rijul Gupta Dec 27 '13 at 18:32

En electrical field induced by a time-varying magnetic field is non-conservative as $$\nabla \times \vec E =-\frac { \partial \vec B}{\partial t}\neq 0$$ It is impossible to derive a potential function for this field due to the non-vanishing rotor. The work done by the force on a charged particle when moving along a closed trajectory can be non-zero. This is not a violation of the energy conservation, as the varying magnetic field implies that energy is supplied to/drained from the field.

How about this example: You have a balloon on a tether. During the day it goes up spinning a dynamo connected to a battery, so it produces mechanical work stored as energy in the battery. During the night you reverse the dynamo and pull the balloon back to the ground, but since the balloon is not heated by the sunlight the buoyancy force is smaller at night, so there is a net gain of energy, you can make a solar power plant this way. The buoyancy force in this case is not conservative but there is no dissipation in the system. It all works because of an external source of energy - same as the loop voltage in the secondary winding of a transformer (or in a synchrotron) when you maintain the time-varying magnetic flux dB/dt via an external energy source. In the latter case, the induced electric field would be a non-conservative and non-dissipative force too.

• Can you please explain a little more on how there was storage of energy and how the external source gave that energy ? – Rijul Gupta Dec 27 '13 at 18:21
• I just put more details in the body of the answer. The ideas of a solar balloon power plant have been around for a while, and looks like inventors have worked out technical details, see, e.g., cleantechnica.com/2009/01/26/… – Maxim Umansky Dec 27 '13 at 18:30
• Granted that there is net work done in this example, but the work done by buoyancy in both the cases is dissipative, it does not matter whether buoyancy is less or more, it is a resistive force and while resisting motion it dissipates energy in both journeys, although the example is good, it does not involve non dissipative non conservative forces – Rijul Gupta Dec 27 '13 at 18:40
• The buoyancy force in this example is non-dissipative if the ascent and descent is done adiabatically (i.e. very slowly). All mechanical work done by/against the buoyancy force here goes to a reversible battery (or other reversible energy storage), and no heat is produced. If there is no external energy source (sunlight) then you could through the cycle indefinitely without any net energy loss or gain. – Maxim Umansky Dec 27 '13 at 20:53

Yes，there are.For example,magnetic force. Assume a magnet walk in a circle alone the magnetic field line,then we know that magnetic force will do work on it.So it couldn't be a conservative force. Also,the work is positive work, so it's non-dissipative force. So we can see,magnetic force is non dissipative-non conservative force.

• Where does this work come from ? How will you justify Energy conservation ? – Rijul Gupta Dec 27 '13 at 17:20
• @rijulgupta This is wrong, magnetic field does no work. Magnetic force is not non-conservative, neither conservative, because the expression conservative is defined only for velocity independent forces (there is no magnetic force vector field). Still, the magnetic field conserves energy (in that sense, it's conservative). – jinawee Dec 27 '13 at 17:51
• But the example in the answer seems to be perfectly reasonable, if we move a very small magnet along the magnetic line of force(I.e. in a magnetic field loop) then at each point a force acts and accelerates the particle, over the loop it should add up. Where is it going wrong ? – Rijul Gupta Dec 27 '13 at 18:17