Shake, agitate, rock, convulse can mean to cause to move to and fro or up and down with more or less violence.
Shake, the most general of these words, in its specific senses usually retains this basic meaning, but it seldom conveys merely this idea.
Very often its meaning is narrowed but enriched by an implication of the particular intent or purpose of the movement; thus, to shake a rug implies an intent to dislodge dust; to shake a tree, to bring down its fruit; to shake a cocktail, to mix ingredients; to shake hands, to greet or to acknowledge an introduction; to shake one’s fist, to threaten. Even in its extended use shake commonly implies movement, usually physical movement.
Agitate usually carries a much stronger implication of tossing or of violent stirring than shake; it often also suggests a prolongation of the movement. When the recipient of the action is a person agitate connotes emotional disturbance or excitement.
Rock suggests a swinging or swaying motion; it tends to lose the implication of lulling derived from its earliest associations with the movement of a cradle and to emphasize those of upheaving, derived from the violent swaying (as of a ship in a storm or of the earth in an earthquake).
Often, especially in extended use, rock suggests, as shake does not, tottering and peril of falling.
Convulse often implies more violence in the motion than any of the others; it also commonly suggests a pulling to and fro or a wrenching or twisting (as of the body in a paroxysm or of the earth in seismic disturbances).