What is the concept of "sympathetic magic" in mythology?
IF we analyse the principles of thought on which magic is based, they will probably be found to resolve themselves into two: first, that like produces like, or that an effect resembles its cause; and, second, that things which have once been in contact with each other continue to act on each other at a distance after the physical contact has been severed.
In terms of specific examples, one might look at Dionysus. His powers derive from the idea that he is like the grape. The sparagmos is a metaphor for the process of making wine--one tears the grape apart, and the fluid is like blood. You bury the casks in a cave, and after a time, spirits are produced. (In the Christian conception, this becomes formalized in the idea of resurrection in the spirit. The Eucharist is another form of sympathetic magic.) Dionysus' power of resurrection is partly based on this, and partly based on the regeneration of the grapevines in spring. Another power of Dionysus is to induce madness. He has this power because intoxication is a type of madness. Euripides' The Bacchae, which is, at its core, a commemoration of Dionysus' dismemberment and resurrection, demonstrates how sympathetic magic is applied in relation to the people of Thebes, and Pentheus in particular. Pentheus (πενθεύς) literally means "suffering", and I don't think it's much of a leap to take that particular ending, "theus" as a form of "theos" (θεός), which renders his name "divine suffering" which is a precise description of Pentheus' story. Pentheus, who starts out the opposite of Dionysus, slowly becomes like Dionysus as the play progresses, with the ultimate end of his taking the place of Dionysus in the ritual commemoration of the sparagmos.
There are many other examples, but I focused on Dionysus because it provides an extremely robust template for analysis.