On a new affine formulation of Hamiltonian classical field theories Juan Carlos Marrero University of La Laguna, Spain e-mail:jcmarrer@ull.edu.es New Trends on Applied Geometric Mechanics Celebrating Darryl Holm’s 70th July 3-7 2017 ICMAT, Madrid, Spain Work in progress with F Gay-Balmaz and N Mart´ ınez Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
CONGRATULATIONS DARRYL!!!!!!! Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Hamilton equations in Classical Mechanics Some basic constructions on a ffi ne bundles Hamiltonian Classical Field Theories of First Order The problems Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Hamilton equations in Classical Mechanics The phase space: Z = T ⇤ Q or Z = T ⇤ Q / G (a symplectic or a Poisson manifold) The observables: C 1 ( Z ) The Poisson bracket: ] : Ω 1 ( Z ) = Γ ( T ⇤ Z ) ! X ( Z ) = Γ ( TZ ) { ' , } = < d ' , ] ( d ) >, ' , 2 C 1 ( Z ) The Hamiltonian: H 2 C 1 ( Z ) The Hamilton equations: A curve s : I ! Z is a solution of the Hamilton eqs if and only if { ' , H } � s = s ⇤ ( d ' ) , 8 ' 2 C 1 ( Z ) ( , is an integral curve of Hamiltonian vector field X h = ] ( dH )) Remark: ⌫ = dt a volume form ) Ω 1 ( I ) ' C 1 ( I ) Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Some basic constructions on a ffi ne bundles A an a ffi ne space which is modeled over the vector space V ; dimV = r A reference frame on A { O ; e 1 , . . . , e r } , O 2 A , { e 1 , . . . , e r } is a basis of V + A + = A ff ( A , R ) the a ffi ne dual of A : a vector space of dimension r + 1 1 A + 2 A + the constant function equal to 1 { e 0 = 1 A + , e 1 , . . . , e r } the dual basis of A + V ⇤ = Lin ( V , R ) the dual space of V { e 1 , . . . , e r } the dual basis on V ⇤ The canonical projection µ : A + ! V ⇤ V ⇤ ' A + / < 1 A + > ' A + / R . Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems The bi-dual space of A A = ( A + ) ⇤ = Lin ( A + , R ) ˜ A is an a ffi ne subspace of ˜ A of codimension 1 V is a vector subspace of ˜ A of codimension 1 U a vector subspace of V + A / U is an a ffi ne space modelled over the vector space V / U and ( A / U ) + ' U 0 = { ' 2 A + / ' ` 2 U 0 } Remark: All the previous constructions may be extended to a ffi ne bundles Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Vertical lifts to a ffi ne bundles ⌧ : A ! Q an a ffi ne bundle modelled over the vector bundle V a 2 A q , u 2 V q ) u v ( a ) = d dt | t =0 ( a + tu ) 2 V a ⌧ ( q i ) local coordinates on Q , { O ; e ↵ } a reference frame on A + ( q i , a ↵ ) local coordinates on A u = u ↵ e ↵ ( q ) ) u v ( a ) = u ↵ @ @ a ↵ | a Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems A ffi ne bundles and principal R -bundles µ : P ! Q a principal R -bundle ( , it is and a ffi ne bundle of rank 1) R 2 X ( P ) the infinitesimal generator of the R -action The a ffi ne space of sections of µ ! {F 2 C 1 ( P ) / R ( F ) = 1 } , h ! F h Γ ( µ ) F h ( p ) = p � h ( µ ( p )) , 8 p 2 P P ( q i , p ) ) R = @ Q ( q i ) , @ p h ( q i ) = ( q i , � H ( q i )) ) F h ( q i , p ) = p + H ( q i ) The typical example: A an a ffi ne space which is modelled over V + µ : A + ! V ⇤ is a principal R -bundle R ⇥ A + ! A + , ( p , ' ) ! ' + c p Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Hamiltonian Classical Field Theories of first order The configuration bundle: ⇡ : P ! M , dim M = m , dim P = m + n P ( x i , u ↵ ) , M ( x i ) vol = dx 1 ^ · · · ^ dx m = d m x vol a volume form on M ; � vol = @ x 1 ^ · · · ^ @ x m = @ m x � vol the volume m -vector; The evolution space: J 1 ⇡ = [ p 2 P { h p : T ⇡ ( p ) M ! T p P / h p is linear and T p ⇡ � h p = id } h p is a horizontal lift from T ⇡ ( p ) M to T p P ⇡ 1 , 0 : J 1 ⇡ ! P is an a ffi ne bundle, ( x i , u ↵ , u ↵ i ) The corresponding vector bundle ⇡ v 1 , 0 : V ( J 1 ( ⇡ )) ! P V ( J 1 ⇡ ) = ⇡ ⇤ ( T ⇤ M ) ⌦ V ⇡ = [ p 2 P { v p : T ⇡ ( p ) M ! V p ⇡ / v p is linear } Local coordinates: ( x i , u ↵ , u ↵ i ) An important remark: Ehresmann connections r : X ( M ) ! X ( P ) on ⇡ : P ! M $ s : P ! J 1 ⇡ 2 Γ ( J 1 ⇡ ) Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems The extended multimomentum bundle: M ⇡ = ( J 1 ⇡ ) + = A ff ( J 1 ( ⇡ ) , R ) = Λ m 2 ( T ⇤ P ) = { ' 2 Λ m ( T ⇤ P ) / i u i v ( ' ) = 0 , 8 u , v 2 V ( ⇡ ) } ( x i , u ↵ , p , p i ⌫ : M ⇡ ! P a vector bundle, ↵ ) ↵ du ↵ ^ d m � 1 x i , ' = pd m x + p i d m � 1 x i = i ( @ x i ) d m x The restricted multimomentum bundle: M 0 ⇡ = Lin ( ⇡ ⇤ ( T ⇤ M ) ⌦ V ( ⇡ ) , R ) ' Lin ( V ⇡ , ⇡ ⇤ ( Λ m � 1 T ⇤ M )) ⌫ 0 : M 0 ⇡ ! P a vector bundle, ( x i , u ↵ , p i ↵ ) The canonical projection: µ : M ⇡ ! M 0 ⇡ µ ( x i , u ↵ , p , p i ↵ ) = ( x i , u ↵ , p i ↵ ) It is a principal R -bundle and M 0 ⇡ = M ⇡ / R Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems The Hamiltonian section: h 2 Γ ( µ ) Γ ( µ ) = { h : M 0 ⇡ ! M ⇡ / h is smooth and µ � h = id } is an a ffi ne space which is modelled over the vector space C 1 ( M 0 ⇡ ) Hamilton-deDonder-Weyl equations: h 2 Γ ( µ ) and h ( x i , u ↵ , p i ↵ ) = ( x i , u ↵ , � H ( x i , u ↵ , p i ↵ ) , p i ↵ ) s 0 : U ✓ M ! M 0 ⇡ a local section of ⇡ � ⌫ 0 , s 0 ( x i ) = ( x i , u ↵ ( x ) , p i ↵ ( x )) s 0 is a solution of H-deD-W eqs , @ u ↵ @ p i @ x i = @ H @ x i = � @ H ↵ , @ p i @ u ↵ ↵ Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems A more geometric approach (Carinena, Crampin, Ibort 1991 and the Spanish group) The canonical multisymplectic structure on M ⇡ ! M ⇡ = du ↵ ^ dp i ↵ ^ d m � 1 x i � dp ^ d m x 2 Ω m +1 ( M ⇡ ) The hamiltonian section h : M 0 ⇡ ! M ⇡ + ! h = h ⇤ ( ! M ⇡ ) a multisymplectic structure on M 0 ⇡ H-deD-W equations s : M ! M 0 ⇡ a local section of ⇡ � ⌫ 0 : M 0 ⇡ ! M s satisfies the H-deD-W eqs , s ⇤ ( i U ! h ) = 0 , 8 U 2 Γ ( V ( ⇡ � ⌫ 0 )) Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Remark: Local sections of ⇡ � ⌫ 0 : M 0 ⇡ ! M ' integral sections of a connection on ⇡ � ⌫ 0 : M 0 ⇡ ! M A connection r : X ( M ) ! X ( M 0 ⇡ ) on ⇡ � ⌫ 0 : M 0 ⇡ ! M is said to be Hamiltonian if i (( Λ m r )( � vol )) ! h = 0 Hamiltonian connections and H-deD-W eqs The integral sections of a Hamiltonian connection are solutions of the H-deD-W eqs for h Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems Some problems: ! h is not a canonical structure (it depends on h ) Hamilton eqs in Classical Mechanics and in Classical Field theories have di ff erent form: Classical Mechanics: < d ' , X H > � s 0 = < d ' , ] ( dH ) > � s 0 = s ⇤ 0 ( d ' ), for ' 2 C 1 ( Z ) and s 0 : I ✓ R ! Z for a curve on Z 0 ( i U ! h ) = 0, for s 0 : M ! M 0 ⇡ a Classical Field Theories: s ⇤ local section and U 2 Γ ( V ( ⇡ � ⌫ 0 )) h is not a real C 1 -function. So, what is dh ?. Where does take values dh ? Hamiltonian connections always exist. But, they are not unique. However, given a Hamiltonian function H there exists a unique Hamiltonian vector field X H associated with H . Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems A possible solution to the last problem: A suitable equivalence relation r connections on ⇡ � ⌫ 0 : M 0 ⇡ ! M r and ¯ r ⇠ ¯ r , r and ¯ r are Hamiltonian for the same Hamiltonian section h Remark: connections on ⇡ � ⌫ 0 : M 0 ⇡ ! M are sections of the a ffi ne bundle ( ⇡ � ⌫ 0 ) 1 , 0 : J 1 ( ⇡ � ⌫ 0 ) ! M 0 ⇡ . Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
Motivation and problems A nice result 9 U a vector subbundle of V ( J 1 ( ⇡ � ⌫ 0 )) ' ( ⇡ � ⌫ 0 ) ⇤ ( T ⇤ M ) ⌦ V ( ⇡ � ⌫ 0 ) such that the equivalence class [ r ] of the Hamiltonian connections for a Hamiltonian section h is a section of the quotient a ffi ne bundle J 1 ( ⇡ � ⌫ 0 ) / U ! M 0 ⇡ . Moreover, U doesn’t depend on h ! L = U 0 ✓ M ( ⇡ � ⌫ 0 ) = Λ m 2 ( M 0 ⇡ ) is given by L = [ ' 0 2 M 0 ⇡ { � 2 Λ m ' 0 M 0 ⇡ ) ϕ 0 � = ( Λ m � 1 ( T ⇤ ' 0 ( ⇡ � ⌫ 0 )))( ¯ ' 0 ( u )) , 2 ( T ⇤ / i ( ¯ ' 0 ) v for u 2 V ⌫ 0 ( ' 0 ) ⇡ } L = < d m x , du ↵ ^ d m � 1 x i , dp i ↵ ^ d m � 1 x i > ) rank L = m + mn + n Remark: J 1 ( ⇡ � ⌫ 0 ) / U and the vector subbundle L ' ( J 1 ( ⇡ � ⌫ 0 ) / U ) + of M ( ⇡ � ⌫ 0 ) will appear more later. They will play an important role! Juan Carlos Marrero On a new a ffi ne formulation of Hamiltonian classical field theor
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