pdq − H dt

pdq − H dt

VIOREL PETREHUS¸

We prove the existence and uniqueness of a Cartan form for Lagrangians onJ^kY. We restricted to fiber manifoldsπX,Y :Y →X whereX is unidimensional.

AMS 2000 Subject Classification: Primary 57R55, Secondary 53Z05.

Key words: fiber space, manifold of jets, Cartan form.

In this paper we follow the lines of [7], [6], [4] from Lagrangians onJ¹Y orJ²Y to Lagrangians on J^kY. The first part is standard and more general computations can be found in [6], [4], [8]. Our result is that the form θLag

associated with a lagrangian Lag onJ^kY is well defined on the space J^2k−1Y, and is contained in Theorem 3. Many other forms θ with some properties of θLagcan be defined, see [5].This form is important in the study of conservation laws for Lag or in the study of conservative numerical methods for the Euler equations associated with Lag.

The basic space of our computations isJ^kY, the space of k jets of sections of the fiber spaceπXY :Y →X. In the sequelX is the real field and Y is an N+ 1 dimensional manifold. The fibre ofπ_XY :Y →X is diffeomorphic to a given manifoldQ. All manifolds are finite dimensional. Ak jet over x∈X is a class of equivalence of sections ofY defined in a neighborhood ofxsuch that two sections are equivalent iff their Taylor developments agree up to orderk.

The jet corresponding to sections will be denoted by j^ks(x). The definition is independent of the local coordinates onX or Q. If x is a local coordinate onXand

y¹, y², . . . , y^N

are local coordinates on Q, then onJ^kY we use the coordinates

x, y^A, y_i^A

A=1,...,N, i=1,...,k, where for the jet of a section s:X → Y, s(x) =

x, s¹(x), s²(x), . . . , s^N(x)

we have y^A = s^A(x), y^A_i = _d^d_xⁱ_is^A(x).

In what follows y0^A is identical with y^A. The correspondence x → j^ks(x) is a section of J^kY and is denoted by j^ks. By truncation of a k development to a l development (l < k) we get a projection π_JlY,J^kY : J^kY → J^lY. The projective limit of these spaces is denoted byJ^∞Y and the limit of their jets

MATH. REPORTS9(59),4 (2007), 357–368

corresponding to section s at x ∈X is denoted j^∞s(x). For more about jet spaces and the change of coordinates see [10],

A projetable diffeomorphism η of Y is a diffeomorphism ηY : Y → Y such that there exists a diffeomorphismη_X of X such that the diagram

(1)

Y −→^ηY Y

 πX,Y

 πX,Y

X −→^ηX X

is commutative; η can be extended to a diffeomorphism η_JkY of J^kY such that the diagram

j^kY ^ηJkY−→ j^kY



π_J^k−1_Y,J^k_Y π_J^k−1_Y,J^k_Y

... ...

j¹Y ^ηJ−→^1Y j¹Y

 π_Y,J1Y

 π_Y,J1Y

Y −→^ηY Y



π_X,Y π_X,Y

X −→^ηX X

is commutative. The extension is given by

(2) η_JkY

j^ks(x)

=j^k

η_Y ◦s◦η⁻_X¹

(η_X(x)).

Using local coordinates we define the total derivativeDx = _∂x^∂ +y1^A ∂

∂y^A+ y^A2 ∂

∂y^A₁ +· · ·+y^A_k∂y^∂A

k−1 as a function from J^kY to the tangent spaceT J^k−1Y, which sends a pointγ ∈J^kY to a vector tangent toJ^k−1Y atπ_Jk−1Y,J^kY (γ).

More conveniently, we considerD_x as a “vector field” on J^∞Y by extending the preceding definition toD_x= _∂x^∂ +_∞

k=0

_N

A=1y^A_k+1∂y^∂A k .

Any functionf :J^kY →R can be considered as a function also denoted abusively f, defined on any JⁿY (n > k) by the composition JⁿY ^πJkY,JnY−→

J^kY −→^f R and consequently defined on J^∞Y. Analogously, any form ω ∈ ΛJ^kY gives a form on JⁿY (n > k) by the formula π_J^∗nY,J^kYω, so it gives a form onJ^∞Y. We defineD_x,p =D_x◦D_x◦ · · · ◦D_x (p times), which gives for f :J^kY → R, a function Dx,pf = DxDx(. . . Dx(f)) defined on J^k+pY. For

p= 0 we define Dx,0= identity. For a section sof Y we have d

dx

f

x, s^A(x),ds^A(x)

dx , . . . ,d^ks^A(x) dx^k

=

= (D_xf)

x, s^A(x),ds^A(x)

dx , . . . ,d^ks^A(x)

dx^k ,d^k+1s^A(x) dx^k+1

= (D_xf)

j^k+1s(x) . For the one parameter group of diffeomorphisms η_Y^ε : Y → Y which covers the one parameter groupη_X^ε :X →X, its infinitesimal generator is of the form

(3) V =V^x(x) ∂

∂x+V^A(x, y) ∂

∂y^A

(by convention the sum is taken over all values of A from 1 to N). The extended groupη^ε_JkY :J^kY →J^kY has a generator denoted by

(4) j^k(V) =V^x(x) ∂

∂x + N A=1

V^A(x, y) ∂

∂y^A + N A=1

k j=1

V_j^A ∂

∂y^A_j . The components ofj^k(V) are given (see [8]) by

V_j^A=D_x,jV^vA+V^xy^A_j+1, where

(5) V^vA =V^A−y₁^AV^x.

The field V^v =

AV^{vA ∂}_∂yA =

A

V^A−y1^AV^x _∂

∂y^A is called the vertical part ofV. The field j^k(V) can be split as

(6) j^k(V) =j^k(V)^v+j^k(V)^h, where the vertical part is

(7) j^k(V)^v =

A

V^vA ∂

∂y^A + N A=1

k j=1

Dx,jV^vA ∂

∂y^A_j = N A=1

k j=0

Dx,jV^vA ∂

∂y_j^A and the horizontal one is

(8) j^k(V)^h =V^x ∂

∂x+ N A=1

k j=0

V^xy^A_j+1

∂

∂y^A_j =V^xDx. One sees that

j^k(V)^h

j^ks(x) =j^ks_∗(x)

V^x ∂

∂x

.

The field j^k(V) is tangent to J^kY, butj^k(V)^v and j^k(V)^h have extra terms involvingy_k^A₊₁ which cancel each other; j^k(V)^v and j^k(V)^h are always defined on J^∞Y by extending the sumation over j to ∞. To avoid such

technical difficulties as (8) not being tangent to J^kY, we shall consider all fields as functions or forms defined onJ^∞Y. Functions and forms are supposed depending onJ^kY for some finitek depending on that form or function. For a vector field V = V^{x ∂}_∂x+

A_∞

j=0V_j^A, we assume that V^x is defined on J^mY and V_j^Adepends on J^j+mY for a fixedm which depends on V;Dx is an example of such a vector field. In this case, the basic operations of calculus with differential forms: outer product, exterior derivative, Lie derivative, etc.

make sense (see [1]).

The differential forms θ^A₀ = θ^A = dy^A −y₁^Adx, θ_j^A = dy_j^A−y^A_j+1dx, for j ≥ 1 are defined using local coordinates. For any section s we have j^ks^∗

θ^A_j

= 0, andθ^A_j (D_x) = 0. For more on jet spaces see [1], [6], [10].

By a Lagrangian on J^kY we understand a differential form of degree 1 which with any jetγ ∈J^kY associates a value in π^∗_X,JkY

Λ¹X

. Using local coordinates, a Lagrangian is a differential form

(9) Lag =L

x, y^A, y^A₁, . . . , y^A_k dx.

Consider now aC^∞-family of sections s^ε: [a(ε), b(ε)]→Y defined forε in a neighborhood of 0, and letζ = _d^d_ε|_ε=0s^ε. Then if all fits into a coordinate chart, integration by parts gives (s⁰ equals s):

d dε

ε=0

_b(ε)

a(e)

L

j^ks^ε(x) dx= (10a)

= d dε

ε=0

_b(ε)

a(e)

L

x, s^εA(x),ds^εA(x)

dx , . . . ,d^ks^εA(x) dx^k

dx=

= N A=1

k j=0

_b

a (−1)^j d dx^j

∂L

∂y_j^A

j^ks(x) ·ζ^A(x)dx+

(10b)

+L

j^ks(b) b(0)−L

j^ks(a) a(0)+

(10c)

+ N A=1

k j=1

j−1

m=0

(−1)^m d dx^m

∂L

∂y_j^A

j^ks(x) ·d^j−m−1ζ^A dx^j−m−1

^b

a

=

= N A=1

k j=0

_b

a (−1)^jD_x.j ∂L

∂y^A_j

j^2ks(x) ·ζ^A(x)dx+

(10d)

+L

j^ks(b) b(0)−L

j^ks(a) a(0)+

(10e)

+ N A=1

k j=1

j−1

m=0

(−1)^mD_x,m ∂L

∂y^A_j

j^2k−1s(x) ·d^j−m−1ζ^A dx^j−m−1

^b

a.

The variation splits into an integral over [a, b] and a boundary part.

We shall express the variation in terms of the differential form below defined locally onJ^2k−1Y as

θLag=Ldx+ N A=1

k j=1

j−1

m=0

(−1)^mD_x,m ∂L

∂y_j^A·θ_j−m−^A 1 = (11)

=Ldx+ N A=1

k−1

b=0 k−b−1

m=0

(−1)^mD_x,m ∂L

∂y_m^A_+b+1 ·θ^A_b .

Let us look at this variation in a different way, as in [6], [4]. Let M the space of C^∞-applications ϕ : [0,1] → Y and ϕ_X : [0,1] → X with ϕ_X injective such thatϕX =πX,Y ◦ϕ. Thens=ϕ◦ϕ⁻_X¹ is a section ofY defined on [a, b] =ϕ_X([0,1])⊂X and we define the action S by

S(ϕ) = _b

a L

j^ks(x) dx= _b

a L j^k

ϕ◦ϕ⁻_X¹

(x) dx= (12)

= _b

a j^k

ϕ◦ϕ⁻_X¹_∗ Lag.

Letη_Y^ε be a one-parameter group of diffeomorphisms of Y which covers the one parameter groupη_X^ε ofX, and letV be its infinitesimal generator. If ϕ^εdef= η^ε_Y ◦ϕand ϕ^ε_X ^def= η_X^ε ◦ϕ_X, then we have ϕ^ε_X =π_X,Y ◦ϕ^ε. In this case, s^ε =ϕ^ε◦ϕ^ε−_X ¹is a section of Y defined on [a(ε), b(ε)] =ϕ^ε_X([a, b]) and there is defined the action

S(ϕ^ε) = _b(ε)

a(ε)

L

j^ks^ε(x) dx= _b(ε)

a(ε)

L j^k

ϕ^ε◦ϕ^ε−_X ¹

(x) dx=

= _b(ε)

a(ε)

j^k

ϕ^ε◦ϕ^ε−_X ¹_∗ Lag.

The definition of η^ε_JkY as η_J^εkY

j^k

ϕ◦ϕ⁻_X¹

(x) =j^k

η^ε_Y ◦ϕ◦ϕ⁻_X¹◦η^ε−_X ¹

(η^ε_X(x)) =

=j^k

ϕ^ε◦ϕ^ε−_X ¹

(η^ε_X(x)) implies

j^k

ϕ◦ϕ⁻_X¹_∗

◦η_J^ε∗kY =η_X^ε∗◦j^k

ϕ^ε◦ϕ^ε−_X ¹_∗ .

Using this, we get (L is the Lie derivative)

∂

∂ε

ε=0S(ϕ^ε) = ∂

∂ε

ε=0

_b(ε)

a(ε)

j^k

ϕ^ε◦ϕ^ε−_X¹_∗ Lag =

= ∂

∂ε

ε=0

_b

a η^ε∗_X j^k

ϕ^ε◦ϕ^ε−_X ¹_∗ Lag =

= ∂

∂ε

ε=0

_b

a j^k

ϕ◦ϕ⁻_X¹_∗

η^ε∗_JkYLag =

= _b

a j^k

ϕ◦ϕ⁻_X¹_∗

L_jk(V)Lag.

Now, we split the Lie derivative asL_jk(V)=L_jk(V)^h+L_jk(V)^v and obtain

[a,b]

j^k

ϕ◦ϕ⁻_X¹_∗

L_jk(V)^hLag =

[a,b]

j^k

ϕ◦ϕ⁻_X¹_∗

LV^xDxLag = (13a)

=

[a,b]

j^k

ϕ◦ϕ⁻_X¹_∗

L_jk(ϕ◦ϕ⁻¹_X)_∗(V^{x ∂}_∂x)Lag = (13b)

=

[a,b]

L_Vx ∂

∂xj^k

ϕ◦ϕ⁻_X¹_∗ Lag =

=

[a,b]

di_Vx ∂

∂x +i_Vx ∂

∂xd j^k

ϕ◦ϕ⁻_X¹_∗ Lag = (13c)

=

[a,b]di_Vx ∂

∂xL j^k

ϕ◦ϕ⁻_X¹

(x) dx=

= L j^k

ϕ◦ϕ⁻_X¹

(x) ·V^xb

a=L j^ks(b)

b(0)−L

j^ks(a)

a(0) = (13d)

= j^2k−1

ϕ◦ϕ⁻¹_{X ∗}

i_j2k−1(V)^hθLag^b

a= (13e)

=

∂[a,b]

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_jk(V)^hθLag.

So, the first two terms of the boundary part of the variation (10e) come fromj^2k−1(V)^h. Necessarily, the third term of (10e) is equal to_b

aj^k

ϕ◦ϕ⁻_X¹_∗

·

L_j^k(V)^vLag. To connect the third term in (10e) withθLag, we look at the equa- tions

ζ(x) = d dε

ε=0s^ε= d dε

ε=0ϕ^ε◦ϕ^ε−_X ¹(x) = d dε

ε=0

η^ε_Y ◦ϕ◦ϕ⁻_X¹◦η^ε−_X ¹ (x) =

= d dε

ε=0

η_Y^ε ◦s◦η^ε−1_X

(x) =V(s(x))−ds

V^x(x) ∂

∂x

=

=

V^A(s(x))−V^x(x)ds^A(x) dx

∂

∂y^A =V^v(s(x)). Hence

d^j−m−1ζ^A(x)

dx^j−m−1 = d^j−m−1

dx^j−m−1V^vA(s(x)) =Dx,j−m−1V^vA

j^j−m−1s(x)

=

=j^2k−1(V)^vA_j−m−1.

Then the third term of the boundary part of the variation (10e) is N

A=1

k j=1

j−1

m=0

(−1)^mDx,m ∂L

∂y_j^A

j^2k−1s(x) ·j^2k−1(V)^vA_j−m−1^b

a

= N A=1

k j=1

j−1

m=0

(−1)^mD_x,m ∂L

∂y_j^A

j^2k−1s(x) ·θ_j−m−^A ₁

j^2k−1(V)^{v b}

a

= j^2k−1s^∗i_j^2k−1(V)^vθLag^b (14) a

=

∂[a,b]

j^2k−1(s)^∗i_j2k−1(V)^vθLag. (15)

Now, (13e) and (15) yield

∂[a,b]j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j2k−1(V)θLag for the boundary part of the variation. These results may be summarized as

Lemma 1. Let Lag = L

x, y^A, y^A₁, . . . , y^A_k

dx be a Lagrangian defined on J^kY andη^ε_Y :Y →Y a one parameter group of projetable diffeomrphisms of Y with infinitesimal generator V. Let η_J^εkY be the extension of η^ε_Y to J^kY and j^k(V) its infinitesimal generator. Then for ϕ ∈ M, ϕ_X([0,1]) = [a, b], such that the image ofϕis contained into a coordinate chart, the variation of the action functional(12) is given by

∂

∂ε

ε=0S(ϕ^ε) = N A=1

k j=0

_b

a (−1)^jDx,j ∂L

∂y_j^A j^2k

ϕ◦ϕ⁻_X¹

(x) ·V^vA(s(x)) dx+

+

∂[a,b]

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j^2k−1(V)θLag.

The exterior derivative of θLag is given by dθLag=

N A=1

k j=1

j−1

m=0

(−1)^md

D_x,m ∂L

∂y^A_j

∧θ_j−m−1^A −

−^N

A=1

k j=1

j−1

m=0

(−1)^mD_x,m ∂L

∂y^A_j dy_j−m^A ∧dx+ dL∧dx.

Now, we are going to compute j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_WdθLag for a vertical vector fieldW =_N

A=1

_2k−1

j=0 W_j^A_∂y^∂A

j onJ^2k−1Y. We have i_WdθLag=

N A=1

k j=1

j−1

m=0

(−1)^md

D_x,m ∂L

∂y^A_j

(W)·θ_j−m−^A 1−

−^N

A=1

k j=1

j−1

m=0

(−1)^mW_j−m−^A 1·d

Dx,m ∂L

∂y_j^A

−

− N A=1

k j=1

j−1

m=0

(−1)^mDx,m ∂L

∂y_j^A ·W_j−m^A ·dx+ N A=1

k j=0

∂L

∂y^A_j W_j^A·dx.

Taking into account thatj^2k−1

ϕ◦ϕ⁻_X¹_∗

θ^A_j = 0, we get j^2k−1

ϕ◦ϕ⁻_X¹_∗

iWdθLag

(16a)

=−^N

A=1

k j=1

j−1

m=0

(−1)^mW_j−m−^A ₁· d dx^m+1

∂L

∂y_j^A

ϕ◦ϕ⁻_X¹ (x)

dx (16b)

−^N

A=1

k j=1

j−1

m=0

(−1)^mW_j−m^A · d dx^m

∂L

∂y_j^A

ϕ◦ϕ⁻_X¹ (x)

·dx (16c)

+ N A=1

k j=0

∂L

∂y_j^A

ϕ◦ϕ⁻_X¹ (x)

·W_j^A·dx (16d)

= N A=1

∂L

∂y₀^A

ϕ◦ϕ⁻_X¹ (x)

·W0^A+ (16e)

+ N A=1

k j=1

(−1)^j d dx^j

D_x,j ∂L

∂y_j^A

ϕ◦ϕ⁻_X¹ (x)

·W0^Adx

= N A=1



^k

j=0

(−1)^j d dx^j

D_x,j ∂L

∂y_j^A

ϕ◦ϕ⁻_X¹ (x)



·W₀^Adx.

(16f)

Now, Lemma 1 may be reformulated as

Proposition 2. Under the hypotheses of Lemma 1, the variation of the functional S is

∂

∂ε|ε=0S(ϕ^ε) =

[a,b]

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j^2k−1(V)dθLag+ (17)

+

∂[a,b]

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j2k−1(V)θLag.

Proof. Because

j^2k−1

ϕ◦ϕ⁻¹_{X ∗}

i_j2k−1(V)^hdθLag=j^2k−1

ϕ◦ϕ⁻¹_{X ∗}

i_j2k−1(ϕ◦ϕ⁻¹_X )_∗(V^{x ∂}_∂x)dθLag

=i_Vx ∂

∂xdj^2k−1

ϕ◦ϕ⁻_X¹_∗

θLag=i_Vx ∂

∂x0 = 0, by (16) we have

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j^2k−1(V)dθLag=j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j^2k−1(V)^vdθLag

= N A=1

k j=0

(−1)^jDx,j ∂L

∂y^A_j j^2k

ϕ◦ϕ⁻_X¹

(x) ·V_j^vA

ϕ◦ϕ⁻_X¹ dx.

The result now follows from Lemma 1.

Now, we shall prove that the definition (11) of θLag is independent of any coordinate chart on the space J^2k−1Y of jets and formula (17) of the variation holds for anyϕ∈ M.

Theorem3. Let the fibration π_X,Y :Y →X, withX a one-dimensional manifold. Let Lag be a Lagrangian defined on J^kY, which is given in a co- ordinate chart by Lag = L

x, y^A, y^A₁, . . . , y_k^A

dx and let η_Y^ε : Y → Y be a one-parameter group (1) of projetable diffeomrphisms of Y with infinitesimal generator V given by (3). Let η^ε_JkY be the extension (2) of η^ε_Y to J^kY and j^k(V) its infinitesimal generator given by (4), (6), (7), (8). Let on J^2k−1Y the formθLag given locally by (11). Then

a) θLag is well defined on J^2k−1Y;

b)for ϕ∈ M, ϕX([0,1]) = [a, b], the variation of the action functional (12)is given by

∂

∂ε|_ε=0S(ϕ^ε) =

[a,b]

j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j2k−1(V)dθLag+ +

∂[a,b]j^2k−1

ϕ◦ϕ⁻_X¹_∗

i_j^2k−1(V)θLag; c)we have the formula

j^k

ϕ◦ϕ⁻¹_{X ∗} Lag =

j^2k−1

ϕ◦ϕ⁻¹_X ^∗θLag;

d)in a standard coordinate chart for J^2k−1Y, θLag only depends at each point j^2k−1(s)(x) on dx and dy_j^A, for j≤k−1;

e) if W is a vector field on J^2k−1Y, tangent to the fibers of π_Y,J^2k−1_Y, then j^2k

ϕ◦ϕ⁻_X¹_∗

(i_WdθLag) = 0;

f)θLag is uniquely defined by b), c), d), e);

g) the Euler-Lagrange equations

(18)

0≤j≤k

(−1)^j

d dx^j

∂L

∂y^A_j

ϕ◦ϕ⁻_X¹

= 0, A= 1,2, . . . , N are equivalent to

(19) j^2k−1

ϕ◦ϕ⁻_X¹_∗

(iW dθLag) = 0 for any vector fieldW on J^2k−1Y.

Proof. If the image of ϕis contained into a coordinate chart, then b) is Proposition 2, c) follows from (11), d) is evident from the definition ofθLag, e) follows from (16) taking into account thatW₀^A = 0. Assuming again that the image of ϕ is contained into a coordinate chart, let θ be any form with properties b), c), d), e) above. It follows from c) anc d) that θ = Ldx + _N

A=1

_k−1

j=0F_A^jθ^A_j. Let ∆ = θ−θLag. Then ∆ = _N

A=1

_k−1

j=0G^j_Aθ_j^A. From b) we get

[a,b]j^2k−1(s)^∗i_j2k−1(V)d∆ = 0 for any projetable vector field V on Y and any sections= ϕ◦ϕ⁻_X¹. From e) we have i_Wd∆ = 0 for any vertical vector fieldπ_Y,J2k−1Y, whence

(20)

[a,b]

j^2k−1(s)^∗i_Wd∆ = 0 for any sections and any vector field

W =W^x(x) ∂

∂x +W^A(x, y) ∂

∂y^A + N A=1

2k−1 j=1

W_j^A ∂

∂y_j^A,