{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Physics 325 Spring 2011 Problem Session 11 Solutions

Physics 325 Spring 2011 Problem Session 11 Solutions - Î 2...

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

Physics 325 Spring 2011 Discussion 11 April 25, 2011 A bead of mass m is constrained to move on a parabola z = λtx 2 , where λ is a positive constant. There is a uniform gravity g pointing in the - z direction. (a) How many degrees of freedom are there? (b) What is the Lagrangian in terms of generalized coordinates (without constraint)? (c) Find the Hamiltonian. (d) Is the Hamiltonian conserved? Solutions : (a) There is one degree of freedom (bead can move on xz -plane, but constrained to a parabola). (b) Use x as generalized coordinate. z = λtx 2 ˙ z = λx 2 + 2 λtx ˙ x L = 1 2 m ( ˙ x 2 + ˙ z 2 ) - mgz L = 1 2 m ( ˙ x 2 + ( λx 2 + 2 λtx ˙ x ) 2 ) - mgλtx 2 L = 1 2 m (1 + 4 λ 2 t 2 x 2 ) ˙ x 2 + 2 2 t ˙ xx 3 - mgλtx 2 + 1 2 2 x 4 (c) The momentum is p = ∂L

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Î» 2 t 2 x 2 ) Ë™ x + 2 mÎ» 2 tx 3 . Equivalently, Ë™ x = p-2 mÎ» 2 tx 3 m (1 + 4 Î» 2 t 2 x 2 ) . (1) 1 H = p Ë™ x-L H = Â± m (1 + 4 Î» 2 t 2 x 2 ) Ë™ x + 2 mÎ» 2 tx 3 Â² Ë™ x-Â³ 1 2 m (1 + 4 Î» 2 t 2 x 2 ) Ë™ x 2 + 2 mÎ» 2 t Ë™ xx 3 Â´ + mgÎ»tx 2-1 2 mÎ» 2 x 4 H = 1 2 m (1 + 4 Î» 2 t 2 x 2 ) Ë™ x 2 + mgÎ»tx 2-1 2 mÎ» 2 x 4 Use Eqn. (1) to write H in terms of only p , x , and t . H = ( p-2 mÎ» 2 tx 3 ) 2 2 m (1 + 4 Î» 2 t 2 x 2 ) + mgÎ»tx 2-1 2 mÎ» 2 x 4 (d) dH dt =-âˆ‚L âˆ‚t Since L has an explicit time-dependence, the Hamiltonian is not conserved. 2...
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business â€˜17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. Itâ€™s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania â€˜17, Course Hero Intern

• The ability to access any universityâ€™s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLAâ€™s materials to help me move forward and get everything together on time.

Jill Tulane University â€˜16, Course Hero Intern