Called precession, the axis is slowly changing, like the spin axis of a top, due to gravity effects of the sun and moon.
14,000 years ago Vega was the north star, not Polaris. Five thousand years ago, Thuban was the North Star. Five thousand years from now, the North Star will be Alpha Cephei. Seven thousand years after that, it will be Vega, and Vega will be 'fixed' and Polaris will draw a circle in the night sky.
Just to correct a minor flaw, the suns most northerly vertical rays are on June 21, which means that the northern hemisphere is tilted toward, not away from the sun.