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I've been searching a bit on the internet for a mathematical description of a non-conservative system, but I could not find it. I'm looking for a good description.

Wikipedia does not have an article on a conservative system but rather a "conservative vector field", but I don't think it has anything to do with it. Or has it?

Some papers use the term non-conservative system from papers back to the 80s and 90s. I wonder if this term has been fading out and exchanged by some other term.

I found something on wolfram.com that gave me some hits, quote:

A conservative system is a system in which work done by a force is

  1. Independent of the path.

  2. Equal to the difference between the final and initial values of an energy function.

  3. Completely reversible.

Question 1: is a conservative vector field and/or conservative force related to a conservative system?
Question 2: does a non-conservative system not include any of the points in the list from the quote above? I.e.

A non-conservative system is one in which work done by a force is:

  1. Dependent on path
  2. Not Equal to the difference between the final and initial values of an energy function.
  3. Completely irreversible.

Is there any consensus on what definition of a non-conservative system is?

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  • $\begingroup$ Do you know what Div , Curl and Grad represent in vector calculus? Adding these terms to your search would help. $\endgroup$
    – StudyStudy
    Jan 29 '20 at 20:14
  • $\begingroup$ Look at this video: youtube.com/watch?v=N7DAqKuSCsk $\endgroup$ Jan 30 '20 at 8:56
  • $\begingroup$ How do you define the energy function? Fds? $\endgroup$ Jan 30 '20 at 9:06
  • $\begingroup$ Did the video answer all of your questions? Or is anything left to ask? Your last one, maybe? I think the first answer below includes all the answers, though. $\endgroup$ Jan 30 '20 at 9:32
  • $\begingroup$ You can also look here: phys.libretexts.org/Bookshelves/University_Physics/… Maybe a bit too technical but that's only good. If you don't understand you can try to understand. If you do understand that's ok too. A win-win situation! $\endgroup$ Jan 30 '20 at 9:44
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Simply put: a conservative system conserves energy, a nonconservative one doesn't.

In a conservative system:

  • trajectories follow paths of constant energy - i.e., if you start the system with a given configuration and let it evolve according to its dynamics, the configuration (say, a particle's position and momentum) might change with time, but its energy remains constant = is conserved;
  • phase-space volumes are preserved - i.e., any arbitrary chunk of the phase space (a blob of initial configurations in the space of possible configurations) maintains a constant volume as it evolves according to the system dynamics; it might deform and even split as much as it wants, but its total volume won't change.

This second description is a statement of Liouville's theorem for Hamiltonian systems, which leads us (see this question) to yet another description of a conservative system, namely a system whose

  • Hamiltonian is autonomous - i.e., it's a function $H(x,p)$ that doesn't depend on time, but only on the phase-space variables $x$ and $p$.

Notice that a nonautonomous Hamiltonian $H(t,x,p)$ can be used for describing a dissipative (i.e., nonconservative) system, but one most often implicitly implies time independence and uses "conservative" and "Hamiltonian" interchangeably. Notice also that for many systems $H$ is just the system's mechanical energy - in this case, $H$ being independent of time is the same as the system's energy being constant.

For mechanical systems, we can also say that, in a conservative system:

Which brings us to your:

Question 1: is conservative vector field and/or conservative force related to a conservative system?

Yes. First, a conservative force is a particular case of a conservative vector field (see, e.g., Wikipedia and this question). Second, the force has to be conservative in order for it to correspond to a meaningful and time-independent potential energy, which in turn you typically need for defining an autonomous Hamiltonian, i.e., for having a conservative system (see also this question, this, this, this and this). Standard examples of dissipative forces are friction and drag.

As for

Question 2: does a non-conservative system not include any of the points in the list from the quote above? I.e., Is the work in a non-conservative system:

  • Dependent on path
  • Not Equal to the difference between the final and initial values of an energy function.
  • Completely irreversible.

Yes. The first two points are equivalent definitions of nonconservative forces (as shown, e.g., in Wikipedia) and therefore preclude, as described in the answer above to Question 1, the system from being conservative.
And yes again - being nonconservative implies a loss or injection of energy in the system that prevents it from "reverting" - returning to a previous configuration. For instance, a dissipative pendulum starting from rest at $3^\circ$ won't manage to climb back again to $3^\circ$ due to the energy lost to, say, the air drag it's subjected to as it oscillates back and fort.
Also reversibility in the sense of time-reversal symmetry would break for nonconservative systems (see this question and this).


All the above is quite pedestrian - for a more complete and sophisticated take, one can start by checking the sources linked.

In particular - it's important to remark - fundamental forces are conservative, so the dissipative forces we see are emergent phenomena (such as friction arising from electromagnetic interactions), or effective/phenomenological descriptions, or a consequence of considering open systems, etc.

As for the expression "nonconservative system" being found more often in papers from 1980's and 90's, I'd guess it's mostly down to the research topic being more active back then.

And lastly, yes, as it I hope is now clear, there is a relative consensus of what a nonconservative system is, even if it often goes unstated.

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  • $\begingroup$ You've done a good job of researching! But of course, you need to know where to look too! $\endgroup$ Jan 30 '20 at 9:41
  • $\begingroup$ @descheleschilder I wish I did know. :-) I intended to write a quick answer, but in searching for references and checking them over, in the end it took hours. $\endgroup$
    – stafusa
    Jan 30 '20 at 21:15
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  1. A conservative vector field and a conservative force are certainly related to the idea of a conservative system. If the work done moving from point A to point B is independent of the path taken from A to B, then the system has a unique energy at every point in space. The gradient of that energy function is an example of a "conservative vector field" as in the Wikipedia article. The gradient vector at any point in space corresponds to the force acting on a particle at that point.

  2. The simplest definition is "a nonconservative system is any system which is not a conservative system" but maybe that doesn't feel very satisfying to you.

There is a hidden assumption which is true for many physical systems even when they are non-conservative: the work done following a path from A to B is equal and opposite to the work done following the reverse path from B to A.

With that assumption, your points 1. and 2. are different ways of saying the same thing. If the work going from A to B is different for two paths P and Q, then if you go from A to B along path P and back to A along the reverse of path Q, you have two different values for the energy function at the same point A, which means you can not describe the energy by a single-valued function.

Your point 3, "completely irreversible" doesn't really mean anything unless you define what the words mean. A real system may have some reversible and some irreversible properties.

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  • $\begingroup$ on point 3: i mean the opposite of "completely reversible", which is a quote from wolfram.com on the definition of a conservative system. $\endgroup$ Jan 29 '20 at 20:43
  • $\begingroup$ Maybe I should have used the word: incompletely reversible.. $\endgroup$ Jan 29 '20 at 20:47
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In dynamical-systems theory, a system is characterised by a set of differential equations describing how the state of a system evolves over time:

$$\dot{x} = f(x),$$

where $f$ can also be thought of as the phase-space flow. Systems are categorised depending on the average of the divergence of the phase-space flow:

  • $\nabla · f = 0$: conservative systemsLiouville’s theorem (the one from theoretical mechanics) gives us that motion in a conservative force field (i.e., with preserved energy) is conservative dynamics in this sense. Note that preservation of energy here applies to the scope of our model in the system, e.g., if we consider the motion of particles, friction converts kinetic energy to heat thus taking it out of the system. Typical examples of conservative systems in this sense are mechanical systems where friction is neglected, e.g., pendulums or celestial mechanics. However, there are also non-physical systems which are conservative, e.g., the classical Lotka–Volterra model where the conserved quantity can be vaguely thought of as biomass.

  • $\nabla · f < 0$: dissipative systems – Most real systems fall in this category. You get these if you look at motion with friction. Most real systems are dissipative. These systems are the main focus of chaos theory (though conservative systems can also be chaotic). A typical example is the damped pendulum, but also the damped and driven pendulum. Another example is the Lorenz system, which is a very rough model for atmospheric dynamics. Here, energy is constantly fed into the system (heating of the atmosphere by the sun) and dissipated.

  • $\nabla · f > 0$: instable system – In mechanics, you get such a system if you constantly feed energy into the system, but have no friction. An example would be the driven but not damped pendulum, where the amplitude escalates. In reality, such systems are not sustainable for long and thus they are of little interest for dynamical-systems theory (which tends to look at the qualitative long-term behaviour).

Now, to come to your question: Non-conservative systems naturally split into two categories (dissipative and instable), which have completely different properties. It makes sense to generally study each of these categories, but there is little to be said about non-conservative systems in general.

It’s pretty much like non-zero numbers: Apart from the fact that you can divide by them, there is little to be said about them.

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A conservative system of particles is a system in which the forces between all particles are conservative. As the name implies, the total energy of all particles is conserved. See for example this video.

Question 1: is a conservative vector field and/or conservative force related to a conservative system? Question 2: does a non-conservative system not include any of the points in the list from the quote above? I.e.

A non-conservative system is one in which work done by a force is:

Dependent on path Not Equal to the difference between the final and initial values of an energy function. Completely irreversible. Is there any consensus on what definition of a non-conservative system is?

A1) Yes it is. It's written in the definition above what a conservative system is
A2) The friction force (which always gives rise to negative work because the force of friction is always opposite to the displacement) doesn't conserve energy for the particles that constitute the system. Energy is released in the form of heat. And clearly this is path-dependent. If I move an ashtray on the table in a straight line from one point A to another B the energy released is the lowest possible (assuming a uniform table and a constant velocity). When you move the ashtray in an erratic way, the energy released becomes higher than the minimum value.

In the comments, I already gave you some links.

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