Are 戦う{たたかう} and 叩く{たたく} related?
They have a similar pronunciation, and their meanings are close enough (fight vs. hit) that I think they could stem from the same origin.
My theory is that 戦う started off as たたき合う but then the sound blended into たたかう, becoming its own word and branching off into its current meanings. But I have basically no knowledge of etymology at all so I am probably way off the mark.
Are they related? And if so, how?
Etymologically, Shogakukan's 国語【こくご】大辞典【だいじてん】 and 大辞林【だいじりん】 list just the ふ auxiliary derivation. Meanwhile, 大辞泉【だいじせん】 lists both the ふ auxiliary and the 合う possibility as derivations.
Given the not-uncommon pattern of deriving iterative / repetitive / continuative verbs by appending ふ to the 未然形【みぜんけい】, and given the appropriate semantics here (叩く "to hit something" ⇒ 戦う "to be hitting something (repeatedly)" → i.e., "to fight"), the ふ auxiliary derivation seems more likely.
Other verb pairs with similar derivations include:
The original ふ often just became う, but in many other cases, this underwent a further shift into the 已然形【いぜんけい】, changing from ふ to へる to える. This expands the list of pairs to include examples such as:
The key semantic shift is that the bare verb expresses a momentary action or change of state, while the verb + ふ expresses an ongoing or repeated action or state.
Yes they are related.
According to デジタル大辞泉:
辞書を引くと、「［動ワ五（ハ四）］《動詞「たた（叩）く」の未然形に反復継続の助動詞「ふ」の付いたものからとも、『叩き合ふ』の音変化とも》」
Basically there are two theories. The one you suggested: a sound change from 叩き合う. (The dictionary entry mentions 『叩き合ふ』 but now all ふ-ending are う-ending verbs)
The second theory is that it come from the verb 叩く to which is appended the 反復継続 (repetition-continuation) auxiliary ふ. This auxiliary clips to the 未然形 so that means 叩か＋ふ. And ふ would become う in modern Japanese, which leads again to たたかう.
You can find this definition for ふ:
［助動］［は｜ひ｜ふ｜ふ｜へ｜（へ）］動詞の未然形に付く。動作・作用の反復・継続を表す。ずっと…しつづける。よく…している。しきりに…している。
I was in the middle of replying that I don't think so, because I was looking at the Kanji etymology for 戦{いくさ}, for example here and of 戦う on weblio.
I was also starting to discuss that there is another Kanji that can be used for たたく, that is 敲{たた}く.. when I decided to dig more and I found something interesting at this link.
According to this source:
「たたかう」の語源は、「叩く」に反復・継続を意味する接尾語「ふ」がついた「たたかふ」つまり「何度も叩く、叩き続ける」という意味の言葉か、「たたきあふ」つまり「互いに叩く」という意味の言葉ではないかといわれている。いずれにせよ、大阪のおばはんの会話みたいな言葉が元になっていると考えられ、どうりで好戦的な意味あいが強いはずである。
So it really seems that your intuition is good. However, rather than being たたき合う it seems that "ふ" was attached as a suffix at first, and in this small explanation they say that the etymology could come from 「たたかふ」("beating several times"), or「たたきあふ」("beat each other"), with the second being very close to what you were thinking.
Anyway, as explained above it seems that this source does not provide a definite answer but clearly adds credit to your guess.
To make the lengthy and too academic story short, 「戦う」and「叩く」are related in its origin and meaning long, long, and long time ago, but are different words, as we use different characters today.
「新明解国語辞典」edited by Kindaichi kyosuke and published by Sanseido defines 戦う as:
[Root] Originally meant 叩き合う. ① to try to conquer the opponent by resorting to (physical or military) forces in order for resolving the dispute. ② to fight for dominance over the rivals in sports and games,
and 「叩く」as:
① beat sth repeatedly. ②beat / tap, clap (hands). ③ attack to beat the enemy compretely. ④ beat (the price) down.
