HW13_prob1(b)_GPS03_G21

# HW13_prob1(b)_GPS03_G21 - After substituting Eq(6 into Eq(2...

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Goldstein 8-21 (3 rd ed. 8.3) We want to produce a new “Hamiltonian”, or “Gamiltonian” G ( · q, · p, t ) ,in which the quantities n · q, · p o are the natural independent variables. We can do this with the following Legendre transform of the Lagrangian L ( q, · q,t ) , G ( · q, · p, t )= X j · p j q j L ( q, · q,t ) , (1) which formally converts the q dependence of L into · p dependence of G .T o f nd the EOM in terms of G we take the total di f erential of Eq.(1) dG = X j h q j d · p j + · p j dq j i dL , (2) where dL = X j " L q j dq j + L · q j d · q j # + L t dt . (3) After using the de f nition of the canonical generalized momentum p j = L · q j , (4) and rewriting the Lagrange EOM as · p j = L q j , (5) Eq.(3) becomes dL = X j h · p j dq j + p j d · q j i + L t dt . (6)

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Unformatted text preview: After substituting Eq.(6) into Eq.(2), we f nd dG = X j h q j d Â· p j âˆ’ p j d Â· q j i âˆ’ âˆ‚ L âˆ‚ t dt . (7) Now express the total di f erential of G in terms of its natural variables as dG = X j " âˆ‚ G âˆ‚ Â· p j d Â· p j + âˆ‚ G âˆ‚ Â· q j d Â· q j # + âˆ‚ G âˆ‚ t dt , (8) 1 and compare the coe ï¬ƒ cients of the independent di f erentials in Eqs.(7) and (8) to f nd q j = âˆ‚ G âˆ‚ Â· p j , (9) p j = âˆ’ âˆ‚ G âˆ‚ Â· q j , (10) and âˆ‚ G âˆ‚ t = âˆ’ âˆ‚ L âˆ‚ t . (11) These are the new equations of motion. 2...
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HW13_prob1(b)_GPS03_G21 - After substituting Eq(6 into Eq(2...

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