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In the following $\left\{x\right\}$ means a configuration point in $3N\text{-dimensional}$ configuration space. Each $x_i$ represents one coordinate of one particle of the system of $N$ particles.

In The Theoretical Minimum: What You Need to Know to Start Doing Physics By Leonard Susskind, George Hrabovsky, Lecture 5, p. 100, Equation (5), the reader is told:

It is a basic mathematical expression of one of the most important principles of physics:

For any system there exists a potential $V\left(\left\{x\right\}\right)$ such that $$F_{i}=\frac{\partial{V\left(\left\{x\right\}\right)}}{\partial{x_i}}.\tag{5}$$

I don't find that assertion objectionable, but it seems like the type of statement which, had I made it, I would be told that I don't know what I'm talking about.

Quite often we find discussions which separate forces expressible as the gradient of a potential from "non-conservative" forces. But that may be more a matter of convenience than a valid segregation of forces in Nature.

Is the above quoted statement a generally accepted principle of physics?


I said that I don't find the quoted statement objectionable; but that is not entirely true. It implies that Nature is fundamentally divisible into particle and void. An assumption that I do not make. But this is the realm of metaphysics.

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    $\begingroup$ Without more context, this is just not a correct mathematical statement. $\endgroup$ – Javier Sep 10 '18 at 13:39
  • $\begingroup$ How so? What part do you have a question about? The mathematical expression is rather typical to me. $\endgroup$ – Steven Thomas Hatton Sep 10 '18 at 13:43
  • $\begingroup$ But like you say, not every vector field is conservative. It is correctly written, but it's false. $\endgroup$ – Javier Sep 10 '18 at 13:53
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    $\begingroup$ He probably means that every fundamental force is conservative. Nonconservative forces only appear if there's stuff going on you're not accounting for, like energy dissipating into heat. $\endgroup$ – knzhou Sep 11 '18 at 7:57
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    $\begingroup$ General tip: Let's not have posts look like revision histories $\endgroup$ – Qmechanic Sep 11 '18 at 8:15
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It should be noted that Susskind & Hrabovsky a few paragraphs before OP's quote write the following disclaimers:

It is quite possible to imagine force laws that do not come from differentiating a potential energy function, but nature does not make use of such non-conservative forces.

and

It is generally not true that if you have a set of functions $F_i(\{x\})$, that they can all be derived by differentiating a single function $V(\{x\})$. It would be a brand-new principle if we asserted that the components of force can be described as (partial) derivatives of a single potential energy function.

It seems that Susskind & Hrabovsky merely try to convey that conservative forces & potential energy are useful principles, especially for fundamental forces and systems where energy is conserved, not that one cannot construct counterexamples.

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Since force is dimensionally the gradient of energy, the statement is correct if F denotes a force and V denotes an energy. Usually V denotes a potential, in which case F should mean some kind of field. Also, F should be a tensor of one rank less than V.

Inconclusion, this statement is vacuous.

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