Is the $\Sigma$ in Newton's second law the sum operator or an "arbitrary" notation?

Question

In high school physics, I often saw the equation: $$\Sigma\vec{F}=m\vec{a}$$At the time, I understood it as "the net force is the sum of all forces acting on a body." Now that I’m studying mathematics at university, I’ve been thinking more formally about vectors. Since $\vec{F}$ is a vector then the sum of all the forces should formally be notated as: $$\sum\vec{F}$$ With the sum operator instead of the greek letter $\Sigma$.

Q: Is $\sum\vec{F}$ equivalent to $\Sigma \vec{F} $? Or is it just a notation that is used for simplicity's sake?

Answer 1

Newton’s law, correctly stated with a summation, is

$$\sum_{i=0}^{n-1}F_i=ma$$

for mass $m$, acceleration of the system $a$, and all $n$ forces considered $F_0,F_1,…,F_{n-1}$.

I don’t know why someone would type a regular uppercase sigma instead of a summation symbol but in any case that’s probably what they meant (a summation over all forces). MathJax has a command \sum that does all the fancy superscripts and subscripts for you, but if you saw something like $\Sigma F$ the author likely meant a sum and didn’t have access to MathJax or LaTeX or the like and had to deal with a regular sigma.

Beware, though, because in some advanced physics (like Kerr black hole stuff) people sometimes use $\sigma$ as an abbreviation for other expressions instead of as an operator. In that context you’d almost always see the full-size summation symbol anyway, so it’s not a huge worry but I thought I’d tack it on anyway.

Answer 2

This is not a physics question, it's a notation question. In most books, you'll find

$$\sum \vec{F} =m\vec{a}, $$

where $\sum$ is the summation symbol. The expression

$$\Sigma \vec{F}=m\vec{a} $$

using "\Sigma" instead of "\sum" is sometimes also used, especially when referring to the equation within a paragraph (as in $\Sigma \vec{F}$). They mean the same thing.

Answer 3

The sum operator $\sum$ is just the greek letter $\Sigma$ (= S in our alphabet) but bigger. It's not an accident. It's just the first S of the word "Sum". It's the same reason why we use a "long S" for the integral sign: $\int$.