DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals wth a eve dmesoal mafold, a phase space, a symplectc structure ad a fucto whch s referred to as the Hamltoa. Usg such a approach, a formulato of mechacs ca be obtaed whch s varat uder group of symplectc dffeomorphsms. Whe formulated wth dfferetal geometrc cocepts, may developmets mechacs ca be smplfed ad uderstood properly. Also may abstract deas of geometry arose the study of mechacs. E.g. the cocept of cotaget budles. The Hamltoa pot of vew allows us to solve completely a seres of mechacal problems whch do ot yeld solutos by other techques.
2. Mathematcal Deftos 2. Topologcal Spaces, Charts ad Mafolds A topologcal space deoted by S has the followg propertes: It has a collecto of subsets, deoted by T. Both the ull set ad S are T. Uo of fte umber of sets T, belogs to T. Itersecto of fte umber of sets T, belogs to T. Ay charts defed o a topologcal space S cossts of a ope set U, together wth oe-oe mappg ϕ: U ϕ( U) R A real dmesoal mafold, M s defed as a space wth fte or coutable collecto of charts such that every pot belogg to ths space s represeted at least oe chart. A N-dmesoal mafold s a space whch s lke R locally. Example: The 2-sphere S = {x Є R 3 x 2 = } s R 2 locally. 2.2 Taget Spaces ad taget budles A taget vector P at a pot P o a mafold M s a ordered par (, P) both of whch are -tuples. may be regarded as a ordary vector ad P the posto vector of the foot of. The set of all possble taget vectors at a pot P o a mafold M s kow as the taget space at P. ( deoted by T P M ) O the mafold M, the collecto of all taget vectors P from pots P M s referred to as the taget budle TM. O smlar les, cotaget vectors/spaces may be defed as dual vectors/spaces of the above. A dual vector beg a lear fuctoal that maps every vector to a member of R. The otato for cotaget to T P M beg T P *M. 2.4 Exteror Forms A -form s a lear fucto ω : R R. ω( λξ + λξ ) = λω( ξ ) + λω( ξ ) 2 2 2 2 2 2 λ, λ R ξ, ξ R
A exteror k-form s a fucto of k vectors whch s k-lear ad atsymmetrc. ' ' ω( λξ + λξ, ξ,... ξ ) = λωξ (,... ξ ) + λωξ (,... ξ ) 2 2 k k 2 k The exteror product ω^ ω 2 o par of vectors ξ^ ξ2 R s the oreted area of the mage of the parallelogram wth the sdes ω( ξ ) ad ω( ξ 2). ω^ ω2 ( ξ, ξ 2) = ω( ξ) ω2( ξ2) ω( ξ2) ω2( ξ) 2.5 Dfferetal Forms A dfferetal -form s a map from taget budle TM to R. ω : TM R ω( ) R P Example: Let (U,φ) be a local chart of M, φ = (x¹,, x ). Every dfferetal - form ω o U ca be wrtte uquely as ω = adx A dfferetal 2-form s a blear, atsymmetrc map from TM TM to R. ω( x, y) R = Ths ca be expressed uquely as ω = a( x) dxλdx j 2.6 Symplectc Structures A symplectc structure o 2 dmesoal mafold, M s a closed, odegeerate dfferetal 2-form o M such that dω 2 =0 2 ξ 0 η: ω ( ξ, η) 0 Example: I R 2 2, ω = dp^dq. j 2.7 Le Groups ad Algebra A Le group s a group G whch s a dfferetable mafold such that group operatos ad verse operatos are dfferetable. A Le algebra s formed by cosderg ftesmal taget space at the detty of le group G. It follows the Jacob detty.
3. Lagraga Mechacs Cosder a dfferetable mafold M ad ts taget budle TM, Lagraga s defed as a dfferetable map L: TM R. A map γ : R M s a moto Lagraga system ad L s a Lagraga fucto f Φ = ( γ ) L( γ ) dt The most geeralzed formulato of a mechacal system s through the prcple of least acto or Hamlto s prcple. Ths postulates the volvemet of a fucto L. The moto of the system s descrbed by the partcular γ for whch t2 S = L( γ ) dt s mmum. The tegral S s refereed to as the acto. The evoluto of the local coordates q = (q q ) of a pot γ(t) uder moto of Lagraga moto o mafold M satsfes the Lagrage-Euler Equato d L L = 0 dt The Lagraga for a system of partcles wth postos (r, r 2 r 3..r ) s gve by 2 L = mv U( r, r... r) t 2 2 a a Upo applcato of a Legedre trasformato to the Lagraga, the resultat so obtaed s called the Hamltoa. 4. Hamltoa Equatos The Legedre Trasform takes a fucto L: TM R defed over taget budle to a fucto H : T* M R o a cotaget budle. The Lagraga ca be represeted terms of ts depedeces as Lq (, q, t).whe a Legedre trasformato s used to chage varables from the dervatves of co-ordates fucto s called the Hamltoa. q H( p, q,) t = q p L( q, q,) t to the mometa The system of Lagrage s equatos s equvalet to 2 frst order equatos dh dh p =, q = dq dq p ; the resultg
O a symplectc mafold (M, ω²), Hamltoa fucto H s a dfferetable map H : M R 5. Mathematcal Model for a Mechacal System A cofgurato space M = M N s a dfferetable mafold of dmeso N. Physcally N s the degrees of freedom of the system. A symplectc structure o the phase space TM, taget budle of M. A hamltoa structure o the phase space T*M, cotaget budle of M. The Ketc eergy s a dfferetable fucto T o TM, such that T: TM R, T (v) = ½ vv,, v TM A force feld s gve by a -form ω = F dx o M 6. Hamltoa Mechacs: 6. Hamltoa Vector Felds: To each vectorξ, taget to a symplectc mafold (M 2, ω 2 ) at a pot x, we assocate a -form ω ξ o ΤΜ x by ω ξ (η) = ω 2 (η, ξ ) N = η ΤΜ () Cosder a somorphsm I: T* M TM such that I preserves symplectc structure. If H s a fucto o a symplectc mafold M 2, dh s a dfferetal -form o M, ad at every pot there s a taget vector to M assocated wth t, defed above(). H s kow as Hamltoa fucto ad I dh s kow as a hamltoa vector feld. 6.2 Hamltoa phase flows: Cosder H: M 2 R a hamltoa fucto o a symplectc mafold M 2. Assumg that feld I dh correspodg to H gves group of dffeomorphsms t 2 g : Μ Μ 2,the The t g d gx t = IdH At t = 0 dx s called the Hamltoa phase flow.
6.3 Law of coservato of eergy The fucto H s a frst tegral of the hamltoa phase flow wth hamltoa fucto H. Cosder the dervatve of H the drecto of vector η s equal to the value of dh o η. But η = I dh from the defto. the eq.() yelds dh(η ) = ω 2 (η, IdH) = ω 2 (η,η ) = 0 Ths shows that H s a frst tegral of hamltoa phase flow. 6.4 Posso Brackets: Cosder a fucto f(p,q,t), the ts total tme dervatve s gve by df f f f = + ( q' k+ p' k) dt t p Posso bracket of the quattes H ad f s gve by H f H f [H,f] = ( ) p p k k k k k k The fuctos of the dyamcal varables whch rema costat durg the moto of the system are kow as tegrals of the moto. 6.5 Defto of Posso bracket from dfferetal geometrc pot of vew: The posso bracket of the fuctos F ad H gve o a symplectc mafold s the dervatve of the fucto F the drecto of the phase flow wth hamltoa fucto H. d [H, F](x) = F t ( g H ( x )) at t = 0 dt The Jacob Idetty: The posso bracket of three fuctos f, g ad h satsfes the Jacob detty: [f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0 Example: Cosder the agular mometum of a partcle M = r p, [ Μ, Μ ] = - Μ, [ Μ, Μ ] = - Μ, [ Z Y Μ, Μ ] = - Μ z Y Y Z Cosder B ad C the hamltoa felds wth hamltoa fuctos B ad C. The [B, C] vector feld s hamltoa ad ts Hamltoa fucto s [B, C]. k
6.6 Le Algebra of Hamltoa felds: The hamltoa felds form subalgebra of the Le algebra of all vector felds. The frst tegrals of a hamltoa phase flow form a subalgebra of the Le algebra of all fuctos. The Le algebra of hamltoa fuctos ca be mapped oto the Le algebra of hamltoa vector felds. 6.7 Caocal Trasformato A caocal trasformato o a Hamltoa system s a chage of co-ordates that whle ot ecessarly preservg the orgal form of the Hamltoa, matas the relatos of caocal co-ordates: H H q =, p = p (Hamlto s equatos) The codtos that o applyg a geeral trasformato of co-ordates, (, q p,) t ( q, p,) t, the Hamlto equatos reta ther form are derved from substtutg the trasformed co-ordates ad comparg wth the orgal : m = p p m m p = p m Tme depedece of the co-ordates ca be corporated to the Hamltoa formalsm by meas of usg tme a caocal trasformato such as: q () t = q( t+ τ ) p () t = p( t+ τ ) The Hamltoa acqures the form H = p( t+ τ) q( t+ τ) + U( p( t+ τ), q( t+ τ), t) 2 = p () t q () t + U( p (), t q (),) t t 2 More geeral defto of caocal trasformatos: Cosder a dfferetable mappg of the phase space R 2 (p, q) to R 2. The mappg g s sad to be a caocal trasformato, f t preserves 2-form 2 ω = dp Λdq 6.8 Geometrc terpretato of the Hamlto-Jacob equato
) Coservatve systems: Hamltoa H : T* M R ad the real valued acto S: M R. ds : M T * M, x ( xds, x ) The composte of these maps gves Hamlto-Jacob equato o M, H ds = E, above equato mples S S Hq (,..., q,,..., ) = E 2) No-coservatve systems: If H s the hamltoa of a dyamcal system, the S S S + Hq (,..., q,,...,, t) = 0 t 7. Refereces: [] V.I. Arold, Mathematcal Methods of Classcal Mechacs, Sprger-Verlag, 978 [2] Ladau ad Lfshtz, Mechacs, Vol. of theoretcal course, Pergamo Press, 960 [3] Vo Westeholz, Dfferetal Forms Mathematcal Physcs, North- Hollad Publshg Compay, 98