Hamlet's letter to Ophelia in 2.2

Act 2 Scene 2 of Hamlet is perhaps one of the most important scenes in the entire play.  One of the questions that has troubled scholars for years is concerning the rapid and extreme change of Hamlet's character.  Act 1 scene 5 shows him very focused, rigid, and immobile; while in 2.2, we see a poetic beauty in Hamlet that has since been missing.  The letter her wrote, read by Polonius is a prime example.  Hamlet writes, 

“Doubt thou the stars are fire,  
Doubt that the sun doth move,
Doubt truth to be a liar,

  But never doubt I love.

 O dear Ophelia, I am ill at these numbers. I have not art to reckon my groans, but that I love thee best, oh, most best, believe it. Adieu.

  Thine evermore, most dear lady,

  whilst this machine is to him,

    Hamlet.”

 

These lines are some of the most poetically beautiful ever written by Shakespeare, and to come from a character who was a polar opposite two scenes before is a very stark changes that raises many questions. The characters around while Polonius reads the letter simply take it as Hamlet finally snapping - they think he has gone completely insane.  I however think that Hamlet is in full control all the time, and simply uses the letter to show even though he has changed, he knows exactly what he is doing in courting Ophelia