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This is a hypothetical question about "pedagogy". Let's say I am trying to take someone who has just a very small amount of knowledge about Newtonian mechanics and convince them that the Lagrangian formulation of mechanics is a natural thing to try. First of all, to convince them that this is natural, at the very least, I would have to show how one can reproduce Newtonian Mechanics from Lagrangian Mechanics, in particular, Newton's Second Law. However, to do this, I would need to introduce the notion of a potential. However, I am having trouble justifying this.

I would like to be able to say "most forces we encounter in nature are conservative, and hence, we are not losing too much by assuming all of our forces are conservative . . .". However, of the two fundamental forces that come to mind in elementary physics, the gravitational force and the electromagnetic force, only one is conservative. I hardly feel that one out of two is a sufficient justification for the introduction of a potential. But if it is not even natural to assume that our forces always come from a potential, then it certainly won't be natural to define the Lagrangian in a way that reproduces Newton's second law.

Is there a nice way of convincing such a student that, even though the electromagnetic force is not conservative, there are still ways of deriving it from some sort of potential? In particular, the difficulty here is that I do not want to assume they have any specific knowledge of electrodynamics.

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2 Answers 2

OP wrote (v1):

I hardly feel that one out of two is a sufficient justification for the introduction of a potential.

I) Here we would like to point out that there exists a velocity-dependent generalized potential

$$U~=~q(\phi - {\bf v}\cdot {\bf A}) $$

for the Lorentz force

$$ {\bf F}~=~ q({\bf E} + {\bf v}\times {\bf B}) . $$

Here $\phi$ is the scalar EM potential and ${\bf A}$ is the magnetic vector potential. The generalized potential $U$ is related to the force ${\bf F}$ as

$$ {\bf F}~=~\frac{d}{dt} \frac{\partial U}{\partial {\bf v}} - \frac{\partial U}{\partial {\bf r}}, $$

see e.g., Herbert Goldstein, Classical Mechanics, Chapter 1, for details. So the electromagnetic force ${\bf F}$ has a potential in the sense of $U$. The Lagrangian is $L=T-U$.

II) Other examples are fictitious forces (centrifugal, Coriolis, etc.), where there also exist generalized potentials, see e.g., Landau and Lifshitz, Vol.1: Mechanics, $\S$39.

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While the electromagnetic force is not conservative in the strict mathematical sense (that is, it cannot be written as the gradient of a scalar), it is conservative in the sense that it conserves energy. This is more than enough to motivate a quantity for potential, as you need some value to cover the non-kinetic energy. After all, the electromagnetic field can be expressed in terms of potential, it's just that you need a 4-vector as opposed to a scalar.

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