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I was going through geometrical optics as it's a part of my undergrad course ,and I found about the Fermat's principle. The principle was understood but the mathematical equation given for it was not so clear to me.

It was something like:

$$\int_A^B ds = \text{minimum}$$ i.e. $$\delta \int_A^B ds = 0$$

This is the equation applicable for light travelling in a homogeneous medium only .

There is also an equation for heterogeneous medium also but my problem will be solved here itself. What I wanted is that -

There's a $\delta$ in front of the line integral, is it the del operator, because nothing is written in the book I am apparently referring to,also where did and del operator come from in this equation.And how come the L.H.S became zero. One thing I am sure is that $ds$ means a small change in the distance traveled by the light while travelling from $A$ (initial point) to $B$ (final point).

Thomas Fritsch
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PATRICK
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  • The symbol $:\delta:$ is not a factor. It's something like the differential symbol $:\mathrm d:$. But while $:\mathrm d f(x):$ is the infinitesimal differential change of $:f(x):$ from $:x \texttt{ to }x\boldsymbol+\mathrm d x:$ the expression $$ \delta\int\limits_{A}^{B}\mathrm d s $$ is the infinitesimal difference change (Variation) from a path (curve) to a very close adjacent one with the same end points. See Calculus of Variations. – Frobenius Oct 10 '21 at 09:09
  • Not Strictly Related : Why one should follow Snell's law for shortest time?. – Frobenius Oct 10 '21 at 09:10
  • Thank you sir for your attention, but can you please tell me what and from where did delta come in that integration. – PATRICK Oct 10 '21 at 09:51
  • It's not easy to answer your questions in a single answer. You must study Calculus of Variations. But if you want to avoid such a time consuming study I suggest you to take a look in the first two chapters of the book $''$ Classical Mechanics $''$ by Goldstein and especially the second one Variational Principles and Lagrange's Equations. – Frobenius Oct 10 '21 at 11:37
  • It is said that "delta" is a notion used to manipulate the parametric families of varied path,thats it.But what is the mathematical description of "delta" and how is it helping us when we multiple it with such vaired path integrals ,such as the fermat's principle? – PATRICK Oct 11 '21 at 06:02

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