# Susskind & Hrabovsky: For any system $F_i=-\partial_{i}V$?

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.

• Without more context, this is just not a correct mathematical statement. – Javier Sep 10 '18 at 13:39
• How so? What part do you have a question about? The mathematical expression is rather typical to me. – Steven Thomas Hatton Sep 10 '18 at 13:43
• But like you say, not every vector field is conservative. It is correctly written, but it's false. – Javier Sep 10 '18 at 13:53
• 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. – knzhou Sep 11 '18 at 7:57
• – Qmechanic Sep 11 '18 at 8:15

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.

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.