Group Actions and Cohomology in the Calculus of Variations J UHA P - PowerPoint PPT Presentation

Group Actions and Cohomology in the Calculus of Variations J UHA P OHJANPELTO Oregon State and Aalto Universities Focused Research Workshop on Exterior Differential Systems and Lie Theory Fields Institute, Toronto, Canada, December 2013

E XAMPLE : I NTEGRABLE SYSTEMS Potential Kadomtsev-Petviashvili (PKP) equation u tx + 3 2 u x u xx + 1 4 u xxxx + 3 s 2 = ± 1 . 4 s 2 u yy = 0 , Admits an infinite dimensional algebra of distinguished symmetries g PKP involving 5 arbitrary functions of time t . (David, Kamran, Levi, Winternitz, Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra , J. Math. Phys. 27 (1986), 1225–1237.)

E XAMPLE : I NTEGRABLE SYSTEMS Potential Kadomtsev-Petviashvili (PKP) equation u tx + 3 2 u x u xx + 1 4 u xxxx + 3 s 2 = ± 1 . 4 s 2 u yy = 0 , Admits an infinite dimensional algebra of distinguished symmetries g PKP involving 5 arbitrary functions of time t . (David, Kamran, Levi, Winternitz, Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra , J. Math. Phys. 27 (1986), 1225–1237.)

PKP EQUATION The symmetry algebra g PKP is spanned by the vector fields ∂ t + 2 ∂ y + ( 1 3 xf ′ − 2 ∂ x + ( − 1 3 uf ′ + 1 X f = f ∂ 3 yf ′ ∂ 9 s 2 y 2 f ′′ ) ∂ 9 x 2 f ′′ − 4 4 243 y 4 f ′′′′ ) ∂ 27 s 2 xy 2 f ′′′ + ∂ u , Y g = g ∂ ∂ y − 2 3 s 2 yg ′ ∂ ∂ x + ( − 4 9 s 2 xyg ′′ + 8 81 y 3 g ′′′ ) ∂ ∂ u , Z h = h ∂ ∂ x + ( 2 3 xh ′ − 4 9 s 2 y 2 h ′′ ) ∂ ∂ u , W k = yk ∂ U l = l ∂ ∂ u , and ∂ u , where f = f ( t ) , g = g ( t ) , h = h ( t ) , k = k ( t ) and l = l ( t ) are arbitrary smooth functions of t .

PKP EQUATION Locally variational with the Lagrangian L = − 1 2 u t u x − 1 x + 1 xx − 3 4 u 3 8 u 2 8 s 2 u 2 y . But the PKP equation admits no Lagrangian that is invariant under g PKP ! To what extent do these properties characterize the PKP-equation?

PKP EQUATION Locally variational with the Lagrangian L = − 1 2 u t u x − 1 x + 1 xx − 3 4 u 3 8 u 2 8 s 2 u 2 y . But the PKP equation admits no Lagrangian that is invariant under g PKP ! To what extent do these properties characterize the PKP-equation?

E XAMPLE : V ECTOR F IELD T HEORIES One-form A = A b ( x i ) dx b on R m satisfying T a = T a ( x i , A b , A b , i 1 , A b , i 1 i 2 , . . . , A b , i 1 i 2 ··· i k ) = 0 , a = 1 , 2 , . . . , m . S YMMETRIES S 1 : spatial translations x i → x i + a i , ( a i ) ∈ R m . S 2 : Gauge transformations A a ( x i ) → A a ( x i ) + ∂φ ∂ x a ( x i ) , φ ∈ C ∞ ( R m ) . C ONSERVATION LAWS C 1 : There are functions t i j = t i j ( x i , A a , A a , i 1 , A a , i 1 i 2 , . . . , A a , i 1 i 2 ··· i l ) such that, for each j = 1 , 2 , . . . , m , A a , j T a = D i ( t i j ) . C 2 : The divergence of T a vanishes identically, D a T a = 0 .

V ECTOR F IELD T HEORIES T HEOREM (A NDERSON , P.) Suppose that the differential operator T a admits symmetries S 1 , S 2 and conservation laws C 1 , C 2 . Then T a arises from a variational principle, T a = E a ( L ) for some Lagrangian L , if (i) m = 2, and T a is of third order; (ii) m ≥ 3, and T a is of second order; (iii) the functions T a are polynomials of degree at most m in the field variables A a and their derivatives. N ATURAL QUESTION : Can the Lagrangian L be chosen to be invariant under [S1], [S2]?

The goal is to reduce these type of questions into algebraic problems.

V ARIATIONAL B ICOMPLEX Smooth fiber bundle F − − − − → E   � π M Adapted coordinates { ( x 1 , x 2 , . . . , x m , u 1 , u 2 , . . . , u p ) } = { ( x i , u α ) } such that π ( x i , u α ) = ( x i ) .

A local section is a smooth mapping σ : U op ⊂ M → E such that π ◦ σ = id . In adapted coordinates σ ( x 1 , x 2 , . . . , x m ) = ( x 1 , x 2 , . . . , x m , f 1 ( x 1 , x 2 , . . . , x m ) , . . . , f p ( x 1 , x 2 , . . . , x m )) .

I NFINITE JET BUNDLE OF SECTIONS J ∞ ( E ) π ∞ o E π ∞ π M

I NFINITE JET BUNDLE Adapted coordinates = ⇒ locally J ∞ ( E ) ≈ { ( x i , u α , u α x j 1 , u α x j 1 x j 2 , . . . , u α x j 1 x j 2 ··· x jk , . . . ) } . Often write u α x j 1 x j 2 ··· x jk = u α j 1 j 2 ··· j k = u α J , where J = ( j 1 , j 2 , . . . , j k ) , 1 ≤ j l ≤ m , is a multi-index .

C OTANGENT BUNDLE OF J ∞ ( E ) dx 1 , dx 2 , . . . , dx m . Horizontal forms : θ α J = du α J − u α Jk dx k . Contact forms : The space of differential forms Λ ∗ ( J ∞ ( E )) on J ∞ ( E ) splits into a direct sum of spaces of horizontal degree r and vertical (or contact) degree s : � Λ ∗ ( J ∞ ( E )) = Λ r , s ( J ∞ ( E )) . r , s ≥ 0 Here ω ∈ Λ r , s ( J ∞ ( E )) is a finite sum of terms of the form J ) dx k 1 ∧ · · · ∧ dx k r ∧ θ α 1 f ( x i , u α , u α j , . . . , u α L 1 ∧ · · · ∧ θ α s L s .

C OTANGENT BUNDLE OF J ∞ ( E ) dx 1 , dx 2 , . . . , dx m . Horizontal forms : θ α J = du α J − u α Jk dx k . Contact forms : The space of differential forms Λ ∗ ( J ∞ ( E )) on J ∞ ( E ) splits into a direct sum of spaces of horizontal degree r and vertical (or contact) degree s : � Λ ∗ ( J ∞ ( E )) = Λ r , s ( J ∞ ( E )) . r , s ≥ 0 Here ω ∈ Λ r , s ( J ∞ ( E )) is a finite sum of terms of the form J ) dx k 1 ∧ · · · ∧ dx k r ∧ θ α 1 f ( x i , u α , u α j , . . . , u α L 1 ∧ · · · ∧ θ α s L s .

H ORIZONTAL AND V ERTICAL D IFFERENTIALS The horizontal connection generated by the total derivative operators ∂ ∂ ∂ ∂ ∂ x i + u α ∂ u α + u α + u α D i = + · · · i ij 1 ij 1 j 2 ∂ u α ∂ u α j 1 j 1 j 2 is flat = ⇒ The exterior derivative splits as d = d H + d V , where d H : Ω r , s → Ω r + 1 , s , d V : Ω r , s → Ω r , s + 1 .

H ORIZONTAL AND V ERTICAL D IFFERENTIALS The horizontal connection generated by the total derivative operators ∂ ∂ ∂ ∂ ∂ x i + u α ∂ u α + u α + u α D i = + · · · i ij 1 ij 1 j 2 ∂ u α ∂ u α j 1 j 1 j 2 is flat = ⇒ The exterior derivative splits as d = d H + d V , where d H : Ω r , s → Ω r + 1 , s , d V : Ω r , s → Ω r , s + 1 .

H ORIZONTAL AND V ERTICAL D IFFERENTIALS m � d H f ( x i , u α , . . . , u α D j f ( x i , u α , . . . , u α J ) dx j , J ) = j = 1 p � � ∂ f J ) θ β d V f ( x i , u α , . . . , u α ( x i , u α , . . . , u α J ) = K . ∂ u β β = 1 | K |≥ 0 K d 2 = 0 = ⇒ d 2 d 2 H = 0 , V = 0 , d H d V + d V d H = 0 .

H ORIZONTAL AND V ERTICAL D IFFERENTIALS m � d H f ( x i , u α , . . . , u α D j f ( x i , u α , . . . , u α J ) dx j , J ) = j = 1 p � � ∂ f J ) θ β d V f ( x i , u α , . . . , u α ( x i , u α , . . . , u α J ) = K . ∂ u β β = 1 | K |≥ 0 K d 2 = 0 = ⇒ d 2 d 2 H = 0 , V = 0 , d H d V + d V d H = 0 .

d V d V d V d V d H d H d H Λ 0 , 1 Λ 1 , 1 Λ m − 1 , 1 Λ m , 1 0 d V d V d V d V d H d H d H Λ 0 , 0 Λ 1 , 0 Λ m − 1 , 0 Λ m , 0 R π ∗ π ∗ π ∗ π ∗ d d d Λ m Λ 0 Λ 1 Λ m − 1 R M M M M

F UNCTIONAL F ORMS Define � α δ ( i 1 j 1 · · · δ i k ) δ β j k , if | I | = | J | , ∂ I α u β J = 0 , otherwise. α : Λ r , s → Λ r , s − 1 , s ≥ 1, Interior Euler operator F I � | I | + | J | � � F I ( − D ) J ( ∂ IJ α ( ω ) = ω ) . α | I | | J |≥ 0 Integration-by-parts operator I : Λ m , s → Λ m , s , s ≥ 1 , I ( ω ) = 1 s θ α ∧ F α ( ω ) . Spaces of functional s-forms F s = I (Λ m , s ) , s ≥ 1. δ V = I ◦ d V : F s → F s + 1 . Then δ 2 Differentials V = 0.

F UNCTIONAL F ORMS Define � α δ ( i 1 j 1 · · · δ i k ) δ β j k , if | I | = | J | , ∂ I α u β J = 0 , otherwise. α : Λ r , s → Λ r , s − 1 , s ≥ 1, Interior Euler operator F I � | I | + | J | � � F I ( − D ) J ( ∂ IJ α ( ω ) = ω ) . α | I | | J |≥ 0 Integration-by-parts operator I : Λ m , s → Λ m , s , s ≥ 1 , I ( ω ) = 1 s θ α ∧ F α ( ω ) . Spaces of functional s-forms F s = I (Λ m , s ) , s ≥ 1. δ V = I ◦ d V : F s → F s + 1 . Then δ 2 Differentials V = 0.

F REE V ARIATIONAL B ICOMPLEX d V d V d V d V δ V d H d H d H I Λ 0 , 2 Λ 1 , 2 Λ m − 1 , 2 Λ m , 2 F 2 0 d V d V d V d V δ V d H d H d H I 0 Λ 0 , 1 Λ 1 , 1 Λ m − 1 , 1 Λ m , 1 F 1 d V d V d V d V E d H d H d H Λ 0 , 0 Λ 1 , 0 Λ m − 1 , 0 Λ m , 0 R π ∗ π ∗ π ∗ π ∗ d d d Λ 0 Λ 1 Λ m − 1 Λ m R M M M M

E ULER -L AGRANGE C OMPLEX ◮ Columns are locally exact ◮ Interior rows are globally exact! Horizontal homotopy operator � � H ( ω ) = 1 θ α ∧ F Ij h r , s c I D I α ( D j ω )] , s ≥ 1 , s | I |≥ 0 | I | + 1 where c I = n − r + | I | + 1 .

E ULER -L AGRANGE C OMPLEX The edge complex d H d H → Λ 0 , 0 → Λ 1 , 0 R − − − − − − − − − − − − → · · · d H d H δ V δ V δ V → Λ m − 1 , 0 Λ m , 0 F 1 F 2 − − − − − − − − → − − − − → − − − − → − − − − → · · · H Div E is called the Euler-Lagrange complex E ∗ ( J ∞ ( E )) .