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A particle of mass m is confined to a parabolic surface of rotation z = ap², where p V + y". The graviational potential is U = mgz. a) Show the Lagrangian is m (² + j² + p²&³) – mgz subject to the constraint that z = ap², i.e. the particle remains on the surfuce of parabolic rotation. b) Use the constraint equation to eliminate z from the Lagrangian and find the generalized momenta p, and p4. Which, if any, are conserved and why?

A particle of mass m is confined to a parabolic surface of rotation z = ap², where p V + y". The graviational potential is U = mgz. a) Show the Lagrangian is m (² + j² + p²&³) – mgz subject to the constraint that z = ap², i.e. the particle remains on the surfuce of parabolic rotation. b) Use the constraint equation to eliminate z from the Lagrangian and find the generalized momenta p, and p4. Which, if any, are conserved and why?

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Please answer both a and b

Document1 - Microsoft Word (Product Activation Failed) File Home Insert Page Layout References Mailings Review View a ? % Cut A Find - Calibri (Body) - 11 - A A Aa AaBbCcDc AaBbCcDc AaBbC AaBbCc AaB AaBbCcL E Copy a Replace Paste B I U - abe x, x I Normal Change Styles I No Spaci.. Heading 1 Heading 2 Title Subtitle A Select - Format Painter Clipboard Font Paragraph Styles Editing A particle of mass m is confined to a parabolic surface of rotation z = VI + y*. The graviational potential is U = mgz. ap?, where p = a) Show the Lagrangian is m (² + j² + pP&#) – mgz L = ap?, i.e. the particle remains on the surface of parabolic subject to the constraint that z = rotation. b) Use the constraint equation to eliminate z from the Lagrangian and find the generalized momenta p, and på. Which, if any, are conserved and why? W ll 12:33 PM 2021-11-09 Page: 1 of 1 Words: 0 B I E E E 90% e +
Transcribed Image Text:Document1 - Microsoft Word (Product Activation Failed) File Home Insert Page Layout References Mailings Review View a ? % Cut A Find - Calibri (Body) - 11 - A A Aa AaBbCcDc AaBbCcDc AaBbC AaBbCc AaB AaBbCcL E Copy a Replace Paste B I U - abe x, x I Normal Change Styles I No Spaci.. Heading 1 Heading 2 Title Subtitle A Select - Format Painter Clipboard Font Paragraph Styles Editing A particle of mass m is confined to a parabolic surface of rotation z = VI + y*. The graviational potential is U = mgz. ap?, where p = a) Show the Lagrangian is m (² + j² + pP&#) – mgz L = ap?, i.e. the particle remains on the surface of parabolic subject to the constraint that z = rotation. b) Use the constraint equation to eliminate z from the Lagrangian and find the generalized momenta p, and på. Which, if any, are conserved and why? W ll 12:33 PM 2021-11-09 Page: 1 of 1 Words: 0 B I E E E 90% e +
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