Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Geometry of PDEs and Hamiltonian systems Olga Rossi University of Ostrava & La Trobe University, Melbourne Bedlewo, May, 2015
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Lagrangian and Hamiltonian systems in jet bundles unique treatment of general Lagrangian systems time independent and time dependent regular and degenerate first order and higher order mechanics and field theory
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Aldaya–de Azc´ arraga, Carath´ eodory, Cari˜ nena, Crampin, De Donder, de Le´ on et al., Dedecker, Echeverr´ ıa-Enr´ ıques–Mu˜ noz-Lecanda–Rom´ an-Roy, Ferraris–Francaviglia, Forger et al., Garcia–Mu˜ noz, Giachetta–Mangiarotti–Sardanashvily, Goldschmidt–Sternberg, Gotay, Grabowski, Grabowska, Ibort, Kanatchikov, Kastrup, Kol´ aˇ r, Krupka, Krupkov´ a-Rossi, Lepage, Marle, Marsden, Massa–Pagani, McLean–Norris, Marrero, Olver, Saunders, Shadwick, Tulczyjew, Vinogradov, Vitagliano, Weyl, . . .
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem REMINDER π : Y → X smooth X orientable π 1 : J 1 Y → X Mechanics / ODEs Field theory / PDEs λ = L ( t , q i , ˙ q i ) dt λ = L ( x i , y σ , y σ j ) ω 0 Cartan form θ λ = Ldt + ∂ L θ λ = L ω 0 + ∂ L ω σ ∧ ω j q i ω i ∂ y σ ∂ ˙ j contact forms: ω i = dq i − ˙ ω σ = dy σ − y σ q i dt j dx j ω 0 = dx 1 ∧ · · · ∧ dx n ω j = i ∂/∂ x j ω 0
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Euler–Lagrange equations J 1 γ ∗ i ξ d θ λ = 0 for every vertical vector field ξ on J 1 Y second order equations solutions = extremals: sections γ of π : Y → X De Donder–Hamilton equations δ ∗ i ξ d θ λ = 0 for every vertical vector field ξ on J 1 Y first order equations solutions = Hamilton extremals: sections δ of π : J 1 Y → X
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Relationship between Lagrangian and Hamiltonian solutions Hamilton equations = equations for integral sections of an EDS generated by n -forms (a Pfaffian system for ODEs) where ξ runs over vertical fields on J 1 Y D = { i ξ d θ λ } Euler–Lagrange equations = equations for holonomic integral sections of the same EDS prolongations of extremals form a subset in the set of Hamilton extremals
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem regular Lagrangians � ∂ 2 L ∂ 2 L � � � det � = 0 det � = 0 ∂ y σ j ∂ y ν q i ∂ ˙ q j ∂ ˙ k bijection between extremals and Hamilton extremals: Euler–Lagrange and Hamilton equations are equivalent
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Hamilton–Jacobi equation w : Y ⊃ U → J 1 Y jet field embedded section: w ◦ γ = J 1 γ w ∗ d θ λ = 0 w ∗ θ λ = dS field of extremals Van Hove theorem embedded section in w satisfying the Hamilton–Jacobi equation is extremal of λ every extremal of a regular λ can be locally embedded into a field of extremals
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Hamiltonian side: Dualization - proper underlying manifold (in place of T ∗ Q ) - Legendre map MECHANICS fibred manifold π : Y → X , dim X = 1 J 1 Y ( t , q i , ˙ q i ) the first jet bundle of π J † Y the extended dual of the first jet bundle = the manifold of real-valued affine maps on the fibres of J 1 Y ( t , q i , p , p i ) with a choice of a volume element on X J † Y = T ∗ Y symplectic manifold Ω = dp ∧ dt + dp i ∧ dq i
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem J ∗ Y the reduced dual = quotient of J † Y by constant (on fibres ( t , q i , p i ) over Y ) maps quotient map ρ : J † Y → J ∗ Y given a Lagrangian system on J 1 Y , construct a dual Hamiltonian system via Legendre map Legendre map p = − L + ∂ L p i = ∂ L Leg : J 1 Y → J † Y q i q i ˙ q i ∂ ˙ ∂ ˙
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem reduced Legendre map p i = ∂ L leg = ρ ◦ Leg : J 1 Y → J ∗ Y ∂ ˙ q i regular Lagrangian � ∂ 2 L � det � = 0 q i ∂ ˙ q j ∂ ˙ hyperregular Lagrangian - if there is an extended Legendre map Leg defined globally, and such that the corresponding reduced Legendre map leg is a diffeomorphism h : J ∗ Y → J † Y Hamiltonian section leg ∗ h ∗ Ω = d θ λ
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem (quite) straightforward generalization to • higher-order regular Lagrangians in mechanics • classical field theory - regular Lagrangians
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem duality restricted to regular Lagrangians BUT: almost all interesting field Lagrangians are singular :-(( Can the class of variational problems having a dual Hamiltonian description be enlarged? YES, BUT concepts of regularity and Legendre transformation have to be revisited AIM: enlarge the class of regular variational problems (with a proper dual Hamiltonian description) as much as possible
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Variational equations revisited: a no-Lagragian viewpoint MOTIVATION L 1 = u 2 L 2 = u 2 x + u x v y − u y v x x , L 1 ∼ L 2 giving the same Euler–Lagrange expressions L 2 is regular, L 1 is not regular Hamilton equations: δ ∗ i ξ d θ λ 2 = 0 are equivalent with the Euler–Lagrange equations duality! δ ∗ i ξ d θ λ 1 = 0 are not equivalent with the Euler–Lagrange equations no duality! – constrained in the sense of Dirac
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem IDEAS Associate Hamilton equations with the Euler–Lagrange form = with the class of equivalent Lagrangians Extend the Euler–Lagrange form to a (proper!) closed ( n + 1)-form EDS i ξ α ∀ ξ Euler–Lagrange equations – holonomic sections Hamilton equations – all sections regularity = property of α guarantees bijection between solutions (for EDS on J 1 Y )
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem LEPAGE MANIFOLDS differential equations in jet bundles: dynamical forms ( ω σ = dy σ − y σ 1-contact, ω σ -generated j dx j ) E = E σ ω σ ∧ ω 0 E σ = E σ ( x i , y ν , y ν j , y ν jk ) sections γ of π such that E vanishes along J 2 γ are solutions of a system of m second order partial differential equations of the form x i , f ν , ∂ f ν ∂ x i , ∂ 2 f ν � � E σ = 0 ∂ x i ∂ x j where f ν are components of a section, γ = ( x i , f ν ).
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Remind: decomposition of differential forms into contact components for ( n + 1)-forms on J r Y π ∗ r +1 , r α = p 1 α + p 2 α + · · · + p n +1 α DEFINITION Lepage ( n + 1)-form a closed ( n + 1)-form α such that p 1 α is a dynamical form. Lepage manifold of order r a fibered manifold π : Y → X where dim X = n , equipped with a Lepage ( n + 1)-form defined on J r Y .
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Lepage ( n + 1)-forms are closed counterparts of Euler–Lagrange forms (variational equations): THEOREM If α is a Lepage ( n + 1)-form then the dynamical form E = p 1 α is locally variational: in a neighbourhood of every point x ∈ Dom α there exists a Lagrangian L such that E is the Euler–Lagrange form of L . Equations for the dynamical form arising from a Lepage ( n + 1)-form are Euler–Lagrange equations. In what follows - first order case: Lepage ( n + 1)-forms on J 1 Y .
Lagrangian and Hamiltonian systems in jet bundles Variational equations revisited: a no-Lagragian viewpoint A new look at the duality problem Structure of first order Lepage ( n + 1)-forms THEOREM Any Lepage ( n + 1)-form may be written as α = α E + η where η is a closed and at least 2-contact form, and where α E is closed and completely determined by E . THEOREM The restriction of α to a suitably small open set U satisfies α | U = d Θ L + d µ, where Θ L is the Poincar´ e–Cartan form of a first-order Lagrangian defined on U , and µ is a 2-contact n -form.
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