U 3 U the relation HAILTON S PRINCIPLE LAGRANGIAN AND HAILTONIAN DYNAICS 83 P P P p t t t 4 beoes 7-3. P P U R ρ φ R ρ Iφ U P P If we take anles an φ as our eneralize oorinates, the kineti ener an the potential ener of the sste are T 5 Interatin 5 fro an point in the reion to an point in the reion, we fin P U R R ρ os P 6 U U 7 or, equivalentl, U U where is the ass of the sphere an where U at the lowest position of the sphere. I is the oent of inertia of sphere with respet to an iaeter. Sine I 5 ρ, the Laranian beoes 8 Now, fro 3 we have L T U R ρ ρ φ R R ρ os 5 When the sphere is at its lowest position, the points A an B oinie. The onition A B ives the equation of onstraint: 88 f, φ R ρ ρφ 4 Therefore, we have two Larane s equations with one uneterine ultiplier: The potential ener is f L λ os φ os φ U t Therefore, the Laranian is f L λ φ φ L φ φ φφφt os φ φ os φ os φ After substitutin 3 an f R ρ an f φ ρ into 5, we fin fro whih R ρ R ρ λ R ρ φφ φ φ φ φ ρ φ λρ 5 φ φ os φ φ φ Fro 7 we fin λ: or, if we use 4, we have λ ρφ φφ 5φ φ φ φ φ φ os φ φ λ R ρ φ 5 84 The Larane9 equations for fin φ an Substitutin into 6, we the φequation of otion with respet to : are φ φ os φ φ φ φ φ ω φ where ω is the frequen of sall osillations, efine b φ φ os φ φ φ φ φ φ 5 ω 7 R ρ 7-8. 7-4. U v 5 5 6 7 v : t v Substitutin into, we fin : This reues to t 9 7-9. 7-6. 7-6. a ωt O a ωt b ξ 7 b φ 8 For ass : For ass : α a ω t b 3 4 aω os bos os aω ωt ωbt os T Tisk Tbob ξ I bob bob b b Substitutin the oorinates for the bob, we obtain Substituteinto into Substitute T ξ I φ φξ os φ a T T The potential ener is iven b U U Uisk U bob isk bobu ξ α os φ an the result is an the result is relation ξ R to reue the erees of freeo to two, an in aition Now let us use the substitute I R Laranian beoes L T for U the isk. b a ωthe os ω t abω os ω t os b os L T U a ω os ω t abω os ω t os b b os 3 L T U ξ φ φξ φ a ξ α os os φ Larane s equation for ives Larane s 4equation for ives eneralize oorinates are The resultin equations for our two abofωotion os ω t os b abw os ω t b t 3 t abω os ω t os b abw os ω t b ξ α φω b abω ω t os abω os ω t φos bφ α ab os φωt α U the relation Larane s equation for the oorinate r leas to P P P p 4 t r rt Ar α t 5 beoes Larane s equation for the oorinate leas to P P U 5 6 r t Interatin 5 fro an point in the reion to an point in the reion, we fin Sine r is ientifie as the anular oentu, 6 iplies that anular oentu is U P onserve. Now, if we use, we anpwrite 6 as 5 ultiplin 7 b r, we have or, equivalentl, 98 6 8 U A L r r r α α v U v U Ain 8 an, we have v k U U 3 b os T v k b os This proble is the ehanial analo of the of liht upon pas fro a eiu of refration withaifferent optial a ertain optial ensit into a eiu ensit. U the eneralize oorinates iven theωfiure, the in a t b Cartesian oorinates for the isk are ξ os α, ξ α, an for the bob the are φ ξ os α, os φ ξ α. The kineti b os b os ener is iven b This reues to 9 9 98 9 we also have Fro U < r r U U > Sine the fore is relate to the potential b If we onsier the potential ener as a funtion of as above, the Laranian of the partile is U f L r U we fin HAILTON S PRINCIPLE LAGRANGIAN AND HAILTONIAN DYNAICS 89 Therefore, Larane s equations for the oorinates an are A α U r 3 Uα beoes where we let Ur. Therefore, the Laranian 4 Let ushoose hooser, the so that the two reions are ivie bthe thepartile ais: is If we, as oorinates the eneralize oorinates, the kineti ener of T an is onstant. Therefore, Fro 9 we have r v t 3 α Pr P 3 Ar r U U 7 7 r α rr r 3 Ar r U U 8 8 or or 7-. 3 4 5 abω ω t os abω os ω t b abω os ω t b φ ξ os φ α φ 6 a ω ω t os b b a b ω ω t os b S
7-7. A C q h U q an ω t e, the, oorinates of the partile are epresse as fro whih Therefore, the kineti ener of the partile is The potential ener is hos q h os ωt q t ωt h q os h ωt q t os ωt hω ωt qω os ωt q ωt hω os ωt qω ωt qos ωt T Then, the Laranian for the partile is B h ω q ω q hω q 3 ω os ω U h t q t 4 L h ω q ω q h ωt q os ωt hω q 5 Larane s equation for the oorinate is q ω q os ωt 6 The opleentar solution an the partiular solution for 6 are written as so that the eneral solution is U the initial onitions, we have Therefore, an or, q t A os iωt δ 7 qp t os ωt ω qt A os iωt δ os ω t 8 ω q Aos δ ω q iωa δ δ, A ω qt os iωt os ωt ω qt osh ωt os ωt ω ω qt In orer to opute the Hailtonian, we first fin the anonial oentu of q. This is obtaine b Therefore, the Hailtonian beoes so that Solvin 3 for H pq L p q ω h q ω ω ω ω os ω ω q hq h q q h t q t qh H q ω h ω q h ωt q ωt t 9 3 os 4 an substitutin ives Solvin 3 for q an substitutin ives p os ω ω ω 5 H hp q h t q ω t The Hailtonian is therefore ifferent fro the total ener, T U. The ener is not onserve in this proble e the Hailtonian ontains tie epliitl. The partile ains ener fro the ravitational fiel. 7-. Fro the fiure, we an easil write own the Laranian for this sste. The resultin equation of otion for is ω R T ω U R os ω os R 3 The equilibriu positions are foun b finin the values of for whih ω R 4 os Note first that an π are equilibriu, an a thir is efine b the onition To investiate the stabilit of eah of these, epan u ε For π, we have os 5 ω R os ω R os 6 ε ω ε ε ε ω ε R iniatin that it is unstable. For, we have ω ε ω R ε 8 whih is stable if ω < R an unstable if ω > R. When stable, the frequen of sall osillations is ω R. For the final aniate, ω ε 9 ε ω with a frequen of osillations of ω ωr, when it eists. Definin a ritial frequen ω R, we have a stable equilibriu at when ω < ω, an a stable equilibriu at ω ω os when ω ω ω ω ω, respetivel. an 4. The frequenies of sall osillations are then ω ω ω To onstrut the phase iara, we nee the Hailtonian H L whih is not the total ener in this ase. A onvenient paraeter that esribes the trajetor for a partiular value of H is H ω K os ωr ω ω so that we ll en up plottin ω K os ω ω for a partiular value of ω an for various values of K. The results for ω < ω are shown in fiure b, an those for ω > ω are shown in fiure. Note how the oriin turns fro an attrator into a separatri as ω inreases throuh ω. As suh, the sste oul ehibit haoti behavior in the presene of apin. 7
.5 K.5 3 b 3.5 K.5 3 3