hw4.solns - Solutions to Homework #4 February 18, 2008 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions to Homework #4 February 18, 2008 1 a.) The time needed to travel between ( x 1 , y 1 ) and ( x 2 , y 2 ) along y ( x ) is given by T [ y ] = integraldisplay T dt = integraldisplay x 2 x 1 dx dt dx = integraldisplay x 2 x 1 dx ds dx dt ds . (1) Here s is the distance along the curve, given by ds 2 = dx 2 + dy 2 , or ds dx = radicalBig 1 + y ′ ( x ) 2 . Also ds/dt is the velocity v along the curve, and by energy conservation we have ds/dt = radicalBig 2 g ( y 1 − y ). We have assumed here that y 2 < y 1 , so that y 1 − y is positive. This gives T [ y ] = integraldisplay x 2 x 1 dx radicaltp radicalvertex radicalvertex radicalbt 1 + y ′ ( x ) 2 2 g ( y 1 − y ( x )) =: integraldisplay x 2 x 1 L ( y ( x ) , y ′ ( x )) dx. (2) b.) By applying the Euler equation d dx parenleftBigg ∂ L ∂y ′ parenrightBigg = ∂ L ∂y we find that the above time is extremized when y ′′ ( x ) = 1 + y ′ ( x ) 2 2( y 1 − y ( x )) (3) We also know that the “Hamiltonian” is conserved (a constant) since x does not appear explicitly in L . Defining p ( x ) = ∂ L /∂y ′ ( x ) and H = p ( x ) y ′ ( x ) −L ( y ( x ) , y ′ ( x )) we can write H ( y ( x ) , y ′ ( x )) = − 1 radicalBig 2 g ( y 1 − y ( x ))(1 + y ′ ( x ) 2 ) = E, (4) where E is a constant. We can now solve foris a constant....
View Full Document

This homework help was uploaded on 02/24/2008 for the course PHYS 3318 taught by Professor Flanagan during the Spring '08 term at Cornell.

Page1 / 5

hw4.solns - Solutions to Homework #4 February 18, 2008 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online