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We often hear these terms. However, they are often confused to be synonyms, but they are not.

What are the rigorous definitions of them?

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    $\begingroup$ This might be helpful physics.stackexchange.com/q/91359 $\endgroup$ – Eagle Mar 14 at 3:04
  • $\begingroup$ @Natasha Yes I have seen that question, but I could not find any rigorous math definitions of these two terminologies. $\endgroup$ – Chetan Waghela Mar 14 at 6:09
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A dissipative force is a force that transfers energy from the macroscopic degrees of freedom into microscopic ones. For instance, the position and velocity of the center of mass of a block of wood sliding on a surface would correspond to macroscopic degrees of freedom. However, the motion of the individual molecules in the surface or the wooden block are microscopic degrees of freedom.

Then, when the wooden block slides on the surface, its molecules collide with the molecules of the surface because of their slight imperfections. As the molecules collide, the energy in the macroscopic motion of the block gets transferred into the random motion of the molecules both in the block and the surface (this random motion is also called heat). However, you are only able to see the motion of the whole block, and you thus describe its motion as under the influence of a dissipative force.

Every dissipative force is non-conservative, that is, it does not conserve the energy in the degrees of freedom you keep track of. Nevertheless, there are non-conservative forces that are not dissipative. This occurs when there is a macroscopic entity you do not keep track of, which adds or takes away energy from your system.

For instance, you can be describing a ball which is attached to a spring which itself is being periodically pulled up and down by some kind of engine. The force of the spring on the ball will be non-conservative, at least if you are only focusing on the ball as the only degree of freedom you are interested in. This is because it can either take, or add energy to the ball on the spring; its action does not conserve the energy of the ball.

Of course, fundamentally speaking, the distinction between non-conservative, dissipative, microscopic, and macroscopic can be a little bit conventional. In fact, physicists generally believe that if we account for all the degrees of freedom involved, every force is conservative. But it is really useful to have effective descriptions in which non-conservative and/or dissipative forces appear.

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