Involutivity of the Hamilton-Cartan equations of a second-order Lagrangian admitting a first-order Hamiltonian formalism

Involutivity of the Hamilton-Cartan equations of a

second-order Lagrangian admitting a …rst-order

Hamiltonian formalism

Jaime Muñoz-Masqué, CSIC , Spain M. Eugenia Rosado, ETSAM, UPM, Spain

July 2012

Introduction

p: E _!N …bred manifold,v volume form on N

Λ=Lv,L₂C∞(J2E), second-order Lagrangian density onp: E _!N.

Hamilton-Cartan equations ofΛ

Generally there is no equivalence between Euler-Lagrange equations and Hamilton-Cartan equations because the Poincaré-Cartan form of a Lagrangian on J2E is generally de…ned on J3E.

Problem: To classify second-order Lagrangians whose Poincaré-Cartan form is projectable ontoJ2E or even toJ1E.

In this case, we study the involutivity of the Hamilton-Cartan equations.

Preliminaries

Jet bundles

N oriented connected smooth manifold,n =dimN.

p: E _!N …bred manifold over N.

Let pk: JkE _!N be the k-jet bundle of p, with natural projections

p_hk: JkE _!JhE, k h.

Coordinates induced by (xi,yα_{), 1 i n, 1 α m=dimE dimN} onJ2E:

(xi,yα_,yα i ,y(αij)),

where

yα

i (jx2s) =

∂(yα s)

∂xi (x), y

α

(ij)(j

2 xs) =

∂2(yα s) ∂xi_∂xj (x).

Preliminaries

Poincaré-Cartan form

The Poincaré-Cartan form attached toΛis de…ned to be the ordinary

n-form onJ3E given by

ΘΛ = (p23) θ2^ωΛ+Λ,

where θ2 is the second-order structure form onJ2E

and ω_Λ is the Legendre form ofΛ, 8

> < > :

θ2 = dyα y_kαdxk ∂

∂yα + dy

α

h y(αhk)dx

k ∂

∂yα h ,

ω_Λ = ( 1)i 1Li0_α vi dyα+ ( 1)i 1Lijαvi dyjα,

vi =dx1^ ^dxci^ ^dxn and the exterior product is taken with respect to the pairing induced by duality, V(p1) J1_E V (p1)!R.

Preliminaries

Poincaré-Cartan form

In coordinates Θ_Λ is given by ΘΛ= ( 1)i 1 Liα0dy

α_+Lih αdy

α

h ^vi + L yiαLi0α y α

(hi)L

ih α v where ₈ > > < > > :

Lij_α = ₂1_δij ∂L ∂yα

(ij)

,

Li0_α = ∂L ∂y_iα

1 2 δijDj

∂L ∂y₍α_ij) .

Dj denoting the “total derivative” with respect toxj, i.e.,

Dj =

∂

∂xj +

∞

∑

m

∑

yα I+(j)

∂ ∂yα

I .

Preliminaries

Projecting ontoJ2E

It is known that the Poincaré-Cartan form of a second-order Lagrangian is projectable onto J2E if and only if the following system of PDEs holds:

1 2 δib

∂2L

∂yacβ ∂y_ibα

+₂1_δia ∂

2_L

∂y_bcβ ∂yα ia

+ ₂1_δic ∂

2_L

∂y_abβ ∂yα ic

=0,

for all indices 1 a b c n,α,β=1,. . .,m.

R. Durán Díaz, J. Muñoz Masqué, Second-order Lagrangians admiting a second-order Hamilton-Cartan formalism, J. Phys. A: Math. Gen.33 (2000), 6003–6016.

Projecting onto

J

E

p2

1:J2E !J1E admits an a¢ ne bundle structure modelled over

W2 = (p1) S2T N (p1₀) V(p)_!J1E.

Theorem

ΘΛ projects onto J1E if and only if

L=Lij_αyα

(ij)+L0, L

ji α =L

ij α 2C

∞_(J1_E),L

0 2C∞(J1E),

0=2∂L hi β

∂yα a

∂Lai_α

∂y_hβ

∂Lah_α

∂y_iβ ,

∂Lih β ∂yα a = ∂L ia α

∂y_hβ

,

a,h,i =1,. . .,n, andα,β=1,. . .,m. Hence, there exists Li 2C∞(J1E) such that locally,

Lih_α = ∂L

i ∂yα h and ∂L h ∂yα i = ∂L i ∂yα h .

Hamiltonian formalism (1)

For Λ=Lv,L₂C∞(J2E)we have:

dΘ_Λ =_Eα(L)θα^v+ ( 1) i

η₂i(L)^vi,

where η₂i(L)is the 2-contact 2-form given by,

η2i(L) =

∂Li0_α ∂yβθ

α

^θβ+ ∂L

i0 α

∂y_jβ ∂Lij_β

∂yα

!

θα^θ_jβ+

∑

j k

∂Li0_α

∂y₍β_jk)θ

α

^θ₍β_jk)

+

∑

i k l

∂Li0 α

∂y₍β_jkl)θ

α_^

θ₍β_jkl)+∂L

ij α

∂y_kβθ

α j ^θ

β

k +

∑

k l

∂Lij_α

∂y₍β_kl)θ

α j ^θ

β

(kl).

Hence the Hamilton-Cartan equations also characterize critical sections for Λ; i.e., for every p3-vertical vector …eld X onJ3E,

s is an extremal forΛ₍₎(j3s) (iXdΘΛ) =0.

Hamiltonian formalism (2)

IfL₂C∞(J2E)whose Poincaré-Cartan form projects onto J1E, then letting 8 > > < > > :

p_αi =Li0_α ∂L

i

∂yα, 1 α m, 1 i n,

H =L0 y_iαLi0_α

∂Li

∂xi, Hamiltonian function of L one obtains,

dΘ_Λ = ( 1)i 1dp_αi _^dyα

^vi +dH^v. Furthermore, ifd10(piα): V(p

1

0)!R are linearly independent, then

s: N_!E is an extremal forΛ if and only if it satis…es:

8 > > > < > > > :

0= ∂(p

i α j

1_s)

∂xi

∂H

∂yα j

1_{s, 1}

α m,

0= ∂(y

α _s)

∂xi +

∂H ∂pi α

j1s, 1 α m, 1 i n.

Regularity

Theorem

If Λ=Lv is a second-order Lagrangian on E such that,

its Poincaré-Cartan form Θ_Λ projects onto J1E ,

the linear forms d10(piα): V(p 1

0)!R,1 α m,1 i n, are

linearly independent,

then every solution to its H-C equations, is holonomic.

Idea If the linear formsd10(piα)are linearly independent hence

∂pi_α

∂y_hβ !

is non-singular

and (s1) i_∂/∂yα

hdΘΛ =0 implies s

1_=j1_{s with s =p}1 0 s1.

Goal

Let L₂ C∞(J2E) whose Poincaré-Cartan form projects ontoJ1E. Let p0: E0 _!N be the bundle de…ned as follows: p0 =p1,E0 =J1E. Let R_L1 J1E0 be the …rst-order di¤erential system de…ned by the solutions to the Hamilton-Cartan equations of Λ=Lv.

We study the involutivity of R_L1 J1E0.

According to the regularity, for every (local) solutions1 to R_L1 one has

s1=j1s, where the section of p:E _!N de…ned bys =p₀1 s1 is called the zero-order section attached to s1.

Involutivity

The morphism of PDE’s

We consider the morphism of …bred manifolds over E0

ϕL:J1E0 !BN = (p0) m+mn(^nT N),

j_x1s1_{7 !} s1(x); ϕα_L(j_x1s1)v ₁

α m, ϕ i α L(j

1 xs1)v

1 i n 1 α m

de…ned by,

8 < :

ϕα_L(jx1s1)v = (s1) i∂/∂yαdΘΛ (x),

ϕi_{α L}(jx1s1)v = (s1) i∂/∂piαdΘΛ (x).

Involutivity

Hamilton-Cartan equations

One has R_L1=kerϕL, where the morphism ϕL: J1E0 !BN is given in local coordinates by ₍

ϕα_L = p_αi_,i +Hyα,

ϕi_{α L} =y_,α_i +Hpi α. where (xi,yα_,pi

α)is a …bred coordinate system for p0: E0 !N, and (xi,yα_,pi

α;y α

,j,pαi,j)be the induced coordinate system on J1E0.

Involutivity

Quasi-linearity

The mapping ϕL: J1E0 !BN is quasi-linear as there exists a vector-bundle morphism, the symbol of ϕL,

σL: (p0) T N V(p0)!BN,

such that,

ϕL(jx1s¯1) =σL(χ1) +ϕL(jx1s1),

8χ1 2TxN Ve0(p0),8j_x1s12 J1E0, with j_x1s1+χ1 =jx1s¯1, ¯

s1(x) =s1(x) =e0.

If χ1= (dxj)x ftjα(χ1)(∂/∂yα)e0+t_ijα(χ1)(∂/∂pi_α)e0g, then

( yα

,j(jx1s¯1) =tjα(χ1) +y,αj(jx1s1),

pi α,j(j

1

xs¯1) =tijα(χ1) +piα,j(j 1 xs1).

Involutivity

Symbol

As Hyα,H_pi α 2C

∞_(E0_)ands¯1_{(x) =s}1_{(x) =e0, we have}

Hyα(j_x1s¯1) =H_yα(j_x1s1), H_pi α(j

1

xs¯1) =Hpi α(j

1 xs1).

Therefore

(

ϕα_L = pi_α,i+Hyα,

ϕi_{α L} =y_,α_i +Hpi α,

=₎ u

α

σL = ∑nh=1thhα ,

ui_α σL =tiα,

(uα_,ui

α)being the standard coordinates induced by the volume form in

BN = m+mn(^nT N). Hence g1 = kerσL

= _fχ1 2TxN Ve0(p0):∑n_h=1t_hhα (χ1) =0,tiα(χ1) =0g.

Involutivity

Formal integrability

R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, and P. Gri¢ ths, Exterior di¤erential systems, Math. Sci. Res. Inst. Publ., Volume 18, Springer-Verlag, Berlin and New York, 1991.

The PDEs system R_L1 J1(E0) isformally integrable if₈k 0,

p_k1+k: R_L1+k _!R_Lk

are surjective, R_L1+k J1+k(E0)being thek-prolongation ofR_L1.

Theorem

If the following conditions are satis…ed:

1 g2 =ker(_σL)

1 is a vector bundle over RL1, where (σL)1 is the

…rst-prolongation of the symbol of R1 L,

2 p1

0: RL1 !E0 is surjective,

3 there exists a quasi-regular basis for T_x for g1 =ker_σL at u2P, then, the PDEs system R1

L is formally integrable.

Involutivity

First prolongation

The …rst prolongation of ϕL is de…ned by, prol₁(ϕL):J2(E0)!J1BN,

prol₁(ϕL)(jx2s1) =jx1(ϕ j1s1).

In coordinates one has,

8 > > > > > > > < > > > > > > > :

uα _prol

1(ϕL) =ϕα_L = pi_α,i+Hyα,

u_αi prol₁(ϕL) = ϕi_{α L} =y_,α_i +H_pi α,

uα

r prol1(ϕL) = pi_α,ir+Hxr_yα+y_,β_rH_yα_yβ+p j β,rHyα_pj

β ,

ui

α,r prol1(ϕL) =y,αir +Hxr_pi α+y

β

,rHyβ_pi α+p

j β,rHpi

αp j β ,

(uα_,ui α,u

α

r,uiα,r)being the induced coordinates on J1BN.

Involutivity

First prolongation of the symbol

As ϕL:J1(E0)!BN is quasi-linear: σk(ϕL) =σ(prol_k(ϕL)),8k 0. Hence ₈χ2 2S2TxN Ve0(p0),8j_x2s12 J2E0, we have:

σ1(ϕL): (p0) S2T N V(p0)!(p0) T N BN,

σ1(ϕL)(χ2) =prol₁(ϕL)(χ2+jx2s1) prol1(ϕL)(j 2 xs1).

If χ2= ∑j k(dxj)x (dxk)x

n tα

jk(χ2)(∂/∂yα)e0+t_ijkα (χ2)(∂/∂p_αi)e0

o

,

then ₍

urβ σ1(ϕL) = ∑na=1(1+δar)taarβ ,

ui_β,r σ1(ϕL) = (1+δir)tirβ.

Involutivity

Quasi-regular basis (1)

Let (X1,. . .,Xn)be a basis of TxN, with dual basis(w1,. . .,wn)and let

SkT_x,

fX1,...,Xjg be the subspace ofS k_T

x generated by the symmetric productswi1 _wik_{, with j +1 i}

1 ik n. For every

e0 ₂E0 with p0(e0) =x, we set g_k,e0_,fX1,...,Xjg =gk,e0\(SkT_x,

fX1,...,Xjg Ve0(p

0_{)), k =1,2.}

The basis (X1,. . .,Xn)is quasi-regular for g1 ate0 if

dimg2,e0 =dimg1_,e0+ n 1

∑

dimg1,e0_,fX1,...,Xjg.

Involutivity

Quasi-regular basis (2)

As

g1= nχ1 2TxN Ve0(p0):

∑

α

hh(χ1) =0,tiα(χ1) =0

o

,

g2= nχ2 2S2TxN Ve0(p0):

∑

o

,

we conclude

dimg1,e0=m(n2 1), dimg2,e0= 1₂mn(n2+n 2), dimg1,e0,_fX1,...,Xjg=m(n

2 _{jn 1).} Hence

n 1

∑

dimg_1,e0_,(X1,...,Xj) = 1

2m(n 1)(n

2 _{2) =dimg}

2,e0 dimg_1,e0,

thus proving that R_L1 is involutive.

Involutivity

Theorem

IfΛ=Lv is a second-order density on p:E _!N whose P-C form projects onto J1_{E and satis…es the regularity condition, then R}1

L is involutive.

If both p: E _!N and Λ are of class Cω_{, then given}

ξ 2(R_L1)x0, there

exists a Cω _{section s of p de…ned on an open neighbourhood U of x} 0 in N

such that, i) j1 x0(j

1_{s) =ξ, and ii) j}1

x(j1s)2 RL1,8x 2U.

Analytically, given scalars λα, λi_α,λ_jα,λi_α,j such that,

0=λi_α,i Hyα(e₀0), 0=λα_i +Hpi

α(e 0 0),

where e₀0 ₂E_x0₀ with coordinates yα_(e0_{) =}

λα,p_αi(e0) =λi_α, then there

exists a solution s1:U _!E0 to the H-C equations de…ned on a neighbourhood of x0 such that,yα(jx10s

1_{) =}

λα,pi_α(jx10s

1_{) =}

λi_α, yα

,j(jx10s

1_{) =}

λα_j,p_αi_,j(jx10s

1_{) =}

λi_α,j.

Applications to General Relativity

Einstein-Hilbert Lagrangian

pM:M !N bundle of pseudo-Riemannian metrics of a given signature

(n+,n ),n++n =n.

(xi,yjk)coordinate system, whereyjk =ykj are de…ned by,

gx =yij(gx)(dxi)x (dxj)x, 8gx 2(pM) 1(U).

The E-H Lagrangian density is given by

(ΛEH)j2

xg =g

ij_(x)(Rg₎h

ihj(x)vg(x) =LEH(jx2g)vx,

v is the standard volume form,

Rg is the curvature tensor of the Levi-Civita connectionΓg ofg,

vg denotes the Riemannian volume form attached to g. Hence,

LEH j2g =ρ(yij g)(Rg)h_ihj,

where ρ= q

( 1)n _det((y

ab)na,b=1).

Applications to General Relativity

Einstein-Hilbert Lagrangian (local expression)

In local coordinates:

LEH =ρ yacybd yabycd yab,cd+L0,

where

L0 = ₂ρ

∑

∑

Fkl,i;rs,jykl,iyrs,j, Fkl,i;rs,j 2C∞(M).

Hence LEH is an a¢ ne function and its P-C form projects onto J1M. We have:

p_kli =

∑

r s

∂2L0

∂yrs,j∂ykl,i

∂(LEH)ijkl

∂yrs

∂(LEH)ijrs

∂ykl

! yrs,j,

H=

∑

k lr

∑

∂2L0

∂yrs,j∂ykl,i

+ ∂(LEH)

ij kl

∂yrs

!

yrs,jykl,i.

Applications to General Relativity

Theorem

The Hamilton-Cartan equations become

8 > > < > > :

0= ∂(p

i kl j1s)

∂xi

∂H ∂ykl

j1s, 1 k l n,

0= ∂(ykl s)

∂xi +

∂H ∂pi_kl j

1_{s, 1 i n, 1 k l n.}

Theorem We have

(i) The E-H Lagrangian satis…es the regularity condition.

(ii) Given symmetric scalars γi_jk =γ_kji , i,j,k =1,. . .,n, there exists a Ricci-‡at (pseudo-)Riemannian metric of signature(n ,n+)de…ned on a neighbourhood of x0 2N such that, gij(x0) =δij,

(Γg₎i

jk(x0) =γi_jk, for all i,j,k.

R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, P. A. Gri¢ ths, Exterior di¤erential systems, Mathematical Sciences

Research Institute Publications, 18, Springer-Verlag, New York, 1991.

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H. Goldschmidt, S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble)23 (1973), no. 1, 203–267.

J. Muñoz Masqué, An axiomatic characterization of the

Poincaré-Cartan form for second-order variational problems, Lecture Notes in Math. 1139, Springer-Verlag 1985, pp. 74–84.

D. J. Saunders, M. Crampin,On the Legendre map in higher-order …eld theories, J. Phys. A: Math. Gen.23 (1990), 3169–3182.