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I'm fascinated by the similarities between them, which both refer to some kind of action meaning alike, namely "wavering" action, and they are morphologically alike too.

If they are cognates, in my opinion, these similarities may be explained reasonably.

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Have you checked Etymonline? –  Matt Эллен Oct 12 '12 at 11:33
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@MattЭллен I'm less and less impressed with Etymonline. It cribs most of its content from OED, and usually leaves out most of the evidence and interesting bits. cf. tchrist's response below. –  StoneyB Oct 12 '12 at 12:30
    
@StoneyB Excellent point, although for folks without OED access it's a good start as a free, valuable resource. –  Zairja Oct 12 '12 at 14:26
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2 Answers

up vote 2 down vote accepted

The direct answer to your question is “maybe”. They aren’t quite sure, which is interesting considering these are words that are relatively new to English. If you look them up in the OED, they present some theories couched in terms of may and might, and present conflicting evidence.

The OED says of the verb sway that it is properly two different words from the standpoint of its etymology:

  1. ME. sweȝe (14th c.), conjugated strong and weak, also swye, to go, move (cf. ME. forsueie to go astray), may have been a native word orig. of the OE. type *sweʒan, (3 pres. ind. *swiʒeþ), pa. t. *swæʒ, parallel to OE. weʒan to move, carry, weigh, (wiʒeþ), wæʒ, ME. weȝe, occas. wye, pa. t. weȝe, wei(ȝ), wei(e)de. (Cf. also the parallelism of swag and wag, sweight and weight.)

    Formally, sweʒe might also be ad. ONor. sveigja to bend (a bow), swing (a distaff), etc., give way, yield (cf. sveigr switch, twig), causative vb. f. svig-, in svig bend, curve, svigi switch, svigna to give way; but the ME. and ONor. verbs do not agree in sense.
  2. The modern sway dates only from c 1500, and agrees in form and sense with, and appears to be ad., LG. swâjen to be moved hither and thither by the wind (whence Sw. svaja to swing, Da. svaie to move to and fro, G. schwaien, schweien), Dutch zwaaien to swing, wave, walk totteringly, slant, bevel.

The verb swag is now chiefly dialectic. The OED says of its etymology:

The existence of this verb is perh. attested for the 15th cent. in swaggyng (s.v. swagging vbl. sb. note), and in swage v.2 Its immediate source is uncertain, but it is prob. Scandinavian: cf. Norw. dial. svagga and svaga to sway (see sway v. etym.).

The English word might correspond to a Scandinavian form of either type (with -gg- or -g-), according to dialect; cf., on the one hand, nag v. (Norw., Sw. nagga), sag v. (Norw. dial. sagga), wag v. (MSw. wagga); on the other, drag v. (ONor. draga), flag sb.2 (Icel. flag, ONor. flaga), snag (Norw. dial. snag, snage); also Sc. swaw = undulating or swinging motion, and flaw sb.1 (ONor. flaga).

The verb swagger is apparently derived from the verb swag above, and is also quite new, first appearing in Shakespeare. The OED provides a note from a 1598 publication that remarks that:

Swaggering is a new worde amongst them, and rounde headed custome giues it priuiledge with much imitation, being created as it were by a naturall Prosopopeia without etimologie or deriuation.

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So this is good evidence that they are connected via swagger, a back and forth movement. I'd just like to point out that currently the word 'swag' by itself now means 'free stuff one gets as a gift' like in a birthday goodie bag or like a gift basket of fruit and beauty products at a fancy hotel. Also, the 'kids these days' use it for 'cool': "That is so swag, dude". None of these meanings have any connection with back and forth movement. –  Mitch Oct 12 '12 at 15:25
    
People will use words for anything. The first sense is "1. A bulgy bag. Obs." and since the OED puts it here (at the same entry) it -must- be related, even though I see no connection at all. –  Mitch Oct 12 '12 at 16:44
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It is possible that they both originate from the Old Norse word sveigja, whose meaning is quite similar to that of swag and sway. So, yes, they might be cognates.

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given that there are resources that can adequately answer this question, supposition is not needed. –  Matt Эллен Oct 12 '12 at 11:34
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It also might be a phonosemantic assonance, like kl-. I'm travelling and away from my sources at the moment, but I think swerve, sway, swing, swat, swoon, and maybe swill might refer to rotatory motion. On va voir. –  John Lawler Oct 12 '12 at 11:35
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