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The core idea of this root was:
bending, swaying, swinging
It’s still quite visible in German words like schwanken and schwenken and even in Schwanz (tail). Just think of a dog waving it :).
The adjective schwach started out as bendable, from where it is not far to its current meaning weak.
The English etymological sources I am using do not mention this root.
But according to the German sources, it would be the origin of the English words to swing, to sway and possibly also swag.