HW13_prob1(a)_GPS01_G01

# HW13_prob1(a)_GPS01_G01 - Problem 8.1 (a) We can dene the...

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Problem 8.1 (a) We can de f ne the Lagrangian L ( q j , · q j ,t ) as the Legendre transform of the Hamiltonian H ( q j ,p j ,t ) , L = X j p j · q j H, (1) where q j , · q j ,and p j are the j th generalized coordinate, velocity, and momentum, respectively. To f nd the EOM in terms of L we take the total di f erential of Eq.(1) dL = X j h p j d · q j + · q j dp j i dH , (2) where dH = X j · H q j dq j + H p j dp j ¸ + H t dt . (3) After using Hamilton’s canonical equations of motion · q j = H p j , (4) and · p j = H q j , (5) Eq.(3) becomes dH = X j h · p j dq j + · q j dp j i + H t dt . (6) After substituting Eq.(6) into Eq.(2), we f nd dL = X j h p j d · q j + · p j dq j i H t dt . (7) Now express the total di f erential of L in terms of its natural variables as dL = X j " L q j dq j + L · q j d · q j # + L t dt , (8) 1

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and compare the coe cients of the independent di
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## This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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HW13_prob1(a)_GPS01_G01 - Problem 8.1 (a) We can dene the...

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