tilt at windmills

• fight windmills

From a passage in the novel Don Quixote where the eponymous character tilts (i.e. jousts) at windmills that he has mistaken for giants.

tilt at windmills (third-person singular simple present tilts at windmills, present participle tilting at windmills, simple past and past participle tilted at windmills)

1. (intransitive) To attack imaginary enemies.
2. (intransitive, by extension) To go on a wild-goose chase; to persistently engage in a futile activity.

• (go on a wild-goose chase): chase a rainbow

• throw one's cap over the windmill
• mistake windmills for giants