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

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Problem 8.1 (a) We can de fi 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 fi nd the EOM in terms of L we take the total di ff 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 fi nd dL = X j h p j d · q j + · p j dq j i H t dt . (7) Now express the total di ff 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 ff erentials in Eqs.(7) and (8) to fi nd · p j = L q j , (9) p j = L · q j , (10) and
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