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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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- Fall '10
- mechanics, Trigraph, Hamiltonian mechanics, pj, Lagrangian mechanics, pj dqj