Sharif University of Technology - CEDRA By: Professor Ali Meghdari

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Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion of Moion fo Inepenen Se of Genealize Cooinaes Lagangian Euaion of Moion fo Depenen Se of Genealize Cooinaes Hailonian Pinciple Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

Hailon s Pinciples I is an inegal pinciple an consies he configuaion of a syse beween he ie ineval,. Avanages; Dynaics Foulaion is:. Reuce o he evaluaion of a scala efinie inegal,. Cooinae syse inepenen in expessing he inegan. Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

By: Pofesso Ali Meghai Le us consie a syse of paicles. Using D Alebes s Pinciple an he Pinciple of Viual Wok we have:.. Can be wien as: i i i f U o x x f U ] [ ] [ Shaif Univesiy of echnology - CEDRA

By: Pofesso Ali Meghai Recall ha he Kineic Enegy fo a Syse of Paicles is:.3 Vaiaion in Kineic Enegy Subsiue in euaion. Shaif Univesiy of echnology - CEDRA

By: Pofesso Ali Meghai On he ohe han, since Viual Wok is efine as: Subsiuing Euaions.3 an.4 ino euaion., we obain: Inegaing Euaions.5 ove he ie ineval o esuls in:.6 ] [ bu U U f U.4.5 Shaif Univesiy of echnology - CEDRA

U [ bu ].6 U.7 x i x i Vaie Pah Geneal Fo of Hailon s Pinciple x i x i ue Pah Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

U.7 Geneal Fo of Hailon s Pinciple: I saes ha he ue pah followe by he ynaic syse o go fo o is such ha he ie inegal of he su of he viual kineic enegy change an viual wok vanishes when subjece o viual isplaceens fo he ue pah. Hailon s Pinciple can be applie o boh on-holonoic an on-consevaive syses. x i x i Vaie Pah x i x i Shaif Univesiy of echnology - CEDRA ue Pah By: Pofesso Ali Meghai

Special Cases: when foces ae consevaive an he viual wok is elae o he change in poenial enegy V by U = -V, we have: L V Lagangian a scala whee : hen; Euaion.7 L,,, an V V becoes funcion V :.8, If he syse is Holonoic, hen euaion.8 becoes: I L Shaif Univesiy of echnology - CEDRA I L.9 By: Pofesso Ali Meghai

I L I.9 Euaion.9 saes ha he ue pah followe by a consevaive holonoic syse o go fo o is such ha he ie inegal I is exeize. L Poof of Lagange s Euaion fo Hailon s Pinciple: Hailon s Pinciple U.7 Fo Holonoic Syse of -Paicles wih egees of feeo we have: Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

By: Pofesso Ali Meghai,,, = Genealize Cooinaes = { } Space =,,,, = Veco Cooinaes of Paicles hen, he oal Kineic Enegy fo he syse is:,,...,,,,...,,.3 Bu, viual wok one by genealize foces ae: aking he vaiaion of using euaion.3, an noing ha =, we have: Q Q U.3 subsiue in.7 Shaif Univesiy of echnology - CEDRA

By: Pofesso Ali Meghai subsiuing in euaion.3, we have: Inegaing he las e of E..3 by pas, we have: ] [ ] [ Q.3 subsiue in.3 [ = = Shaif Univesiy of echnology - CEDRA

[ Q ].33 Since in Holonoic Syses, he genealize cooinaes fo an inepenen se, heefoe, he coefficiens of each in euaion.33 us be zeo. heefoe: Q,,..., M.34 Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

Exaple: A bea of ass is fee o slie on a hoop of aius R as shown. he hoop is oaing wih he consan angula velociy Ω. Fin he euaion of oion using Hailon s pinciple?. Moion: Le x, x, x 3 be aache o he hoop. g R x 3 Ω x Rsin e Rcos e 3 x v Rsin e R cos e R sin e 3 Dau Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

. Kineic Enegy: v v [ Rsin R R sin R cos R sin 3. Poenial Enegy: aking θ= as he au, we have; V gr cos 4. Lagangian: L V R sin R gr cos Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

5. he Vaiaion of Lagangian: L L L R g [ sin cos sin ] R R o apply Hailon s Pinciple, we nee o expess he n e in above euaion in es of δθ. Inegaing he n e by pas esuls: he inegae e in he above euaion vanishes by efiniion of he vaiaion a he beginning an en of he pah. heefoe, applying Hailon s Pinciple esuls: L [ R R sin cos g R sin ] Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

6. Applying he Hailon s Pinciple: L [ R R sin cos g R sin ] Fo he eualiy o hol, he inegan us vanish a all ies. Because δθ is abiay, fo he inegan o be zeo, he coefficien of δθ us be zeo. heefoe, he Euaion of Moion will esuls as: sin g R cos Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai

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