A Geometric Hamilton-Jacobi Theory for Classical Field Theories

0.17 MB·English

by  M. de Leon

1
A GEOMETRIC HAMILTON-JACOBI THEORY FOR
CLASSICAL FIELD THEORIES
MANUEL DE LEO´N, JUAN CARLOS MARRERO,
AND DAVID MART´IN DE DIEGO
Abstract. In this paper we extend the geometric formalism of
the Hamilton-Jacobitheory for hamiltonianmechanics to the case
8
of classical field theories in the framework of multisymplectic ge-
0
ometry and Ehresmann connections.
0
2

n
a
J

8 Contents

]
1. Introduction 1
h
p
2. A geometric Hamilton-Jacobi theory for Hamiltonian
-
h mechanics 2
t
a 3. The multisymplectic formalism 3
m
3.1. Multisymplectic bundles 3
[
3.2. Ehresmann Connections in the fibration π : J1π∗ −→ M 5
1 1
v 3.3. Hamiltonian sections 5
1
8 3.4. The field equations 5
1
4. The Hamilton-Jacobi theory 6
1
1. 5. Time-dependent mechanics 9
0
References 10
8
0
:
v
i
X
1. Introduction
r
a
The standard formulation of the Hamilton-Jacobi problem is to find
a function S(t,qA) (called the principal function) such that
∂S ∂S
+H(qA, ) = 0. (1.1)
∂t ∂qA
1To Prof. Demeter Krupka in his 65th birthday
2000 Mathematics Subject Classification. 70S05, 49L99.
Key words and phrases. Multisymplectic field theory, Hamilton-Jacobi
equations.
This work has been partially supported by MEC (Spain) Grants MTM 2006-
03322, MTM 2007-62478, project “Ingenio Mathematica” (i-MATH) No. CSD
2006-00032(Consolider-Ingenio 2010) and S-0505/ESP/0158of the CAM.
1
2 M. DE LEO´N,J.C. MARRERO,ANDD. MART´INDE DIEGO
If we put S(t,qA) = W(qA) − tE, where E is a constant, then W
satisfies
∂W
H(qA, ) = E; (1.2)
∂qA
W is called the characteristic function.
Equations (1.1) and (1.2) are indistinctly referred as the Hamilton-
Jacobi equation.
There are some recent attempts to extend this theory for classical
field theories in the framework of the so-called multisymplectic formal-
ism [15, 16]. For a classical field theory the hamiltonian is a function
H = H(xµ,yi,pµ), where (xµ) are coordinates in the space-time, (yi)
i
represent the field coordinates, and (pµ) are the conjugate momenta.
i
In this context, the Hamilton-Jacobi equation is [17]
∂Sµ ∂Sµ
+H(xν,yi, ) = 0 (1.3)
∂xµ ∂yi
where Sµ = Sµ(xν,yj).
In this paper we introduce a geometric version for the Hamilton-
Jacobi theory based in two facts: (1) the recent geometric descrip-
tion for Hamiltonian mechanics developed in [6] (see [8] for the case
of nonholonomic mechanics); (2) the multisymplectic formalism for
classical field theories [3, 4, 5, 7] in terms of Ehresmann connections
[9, 10, 11, 12].
We shall also adopt the convention that a repeated index implies
summation over the range of the index.
2. A geometric Hamilton-Jacobi theory for Hamiltonian
mechanics
First of all, we give a geometric version of the standard Hamilton-
Jacobi theory which will be useful in the sequel.
Let Q be the configuration manifold, and T∗Q its cotangent bundle
equipped with the canonical symplectic form
ω = dqA ∧dp
Q A
where (qA) are coordinates in Q and (qA,p ) are the induced ones in
A
T∗Q.
Let H : T∗Q −→ R a hamiltonian function and X the correspond-
H
ing hamiltonian vector field:
i ω = dH
XH Q
The integral curves of X , (qA(t),p (t)), satisfy the Hamilton equa-
H A
tions:
dqA ∂H dp ∂H
A
= , = −
dt ∂p dt ∂qA
A
HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 3
Theorem 2.1 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form
on Q (that is, dλ = 0 and, locally λ = dW). Then, the following
conditions are equivalent:
(i) If σ : I → Q satisfies the equation
dqA ∂H
=
dt ∂p
A
then λ◦σ is a solution of the Hamilton equations;
(ii) d(H ◦λ) = 0.
To go further in this analysis, define a vector field on Q:
Xλ = Tπ ◦X ◦λ
H Q H
as we can see in the following diagram:
T∗Q XH // T(T∗Q)
??
λ πQ TπQ
(cid:15)(cid:15) Xλ (cid:15)(cid:15)
Q H // TQ
Notice that the following conditions are equivalent:
(i) If σ : I → Q satisfies the equation
dqA ∂H
=
dt ∂p
A
then λ◦σ is a solution of the Hamilton equations;
(i)’ If σ : I → Q is an integral curve of Xλ, then λ◦σ is an integral
H
curve of X ;
H
(i)” X and Xλ are λ-related, i.e.
H H
Tλ(Xλ) = X ◦λ
H H
so that the above theorem can be stated as follows:
Theorem 2.2 (Hamilton-Jacobi Theorem). Let λ be a closed 1-form
on Q. Then, the following conditions are equivalent:
(i) Xλ and X are λ-related;
H H
(ii) d(H ◦λ) = 0.
3. The multisymplectic formalism
3.1. Multisymplectic bundles. The configuration manifold in Me-
chanics is substituted by a fibred manifold
π : E −→ M
4 M. DE LEO´N,J.C. MARRERO,ANDD. MART´INDE DIEGO
such that
(i) dimM = n, dimE = n+m
(ii) M is endowed with a volume form η.
We can choose fibred coordinates (xµ,yi) such that
η = dx1 ∧···∧dxn .
We will use the following useful notations:
dnx = dx1 ∧···∧dxn
dn−1xµ = i dnx .
∂
∂xµ
Denote by Vπ = kerTπ the vertical bundle of π, that is, their ele-
ments are the tangent vectors to E which are π-vertical.
Denote by
Π : ΛnE −→ E
the vector bundle of n-forms on E.
The total space ΛnE is equipped with a canonical n-form Θ:
Θ(α)(X ,...,X ) = α(e)(TΠ(X ),...,TΠ(X ))
1 n 1 n
where X ,...,X ∈ T (ΛnE) and α is an n-form at e ∈ E.
1 n α
The (n+1)-form
Ω = −dΘ ,
is called the canonical multisymplectic form on ΛnE.
Denote by ΛnE the bundle of r-semibasic n-forms on E, say
r
ΛnrE = {α ∈ ΛnE | iv1∧···∧vrα = 0, whenever v1,...,vr are π-vertical}
Since ΛnE is a submanifold of ΛnE it is equipped with a multisym-
r
plectic form Ω , which is just the restriction of Ω.
r
Two bundles of semibasic forms play an special role: ΛnE and ΛnE.
1 2
The elements of these spaces have the following local expressions:
ΛnE : p dnx
1 0
ΛnE : p dnx+pµdyi∧dn−1xµ .
2 0 i
whichpermitstointroducelocalcoordinates(xµ,yi,p )and(xµ,yi,p ,pµ)
0 0 i
in ΛnE and ΛnE, respectively.
1 2
Since ΛnE is a vector subbundle of ΛnE over E, we can obtain the
1 2
quotient vector space denoted by J1π∗ which completes the following
exact sequence of vector bundles:
0 −→ ΛnE −→ ΛnE −→ J1π∗ −→ 0 .
1 2
We denote by π : J1π∗ −→ E and π : J1π∗ −→ M the induced
1,0 1
fibrations.
HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 5
3.2. Ehresmann Connections in the fibration π : J1π∗ −→ M.
1
A connection (in the sense of Ehresmann) in π is a horizontal sub-
1
bundle H which is complementary to Vπ ; namely,
1
T(J1π∗) = H⊕Vπ
1
where Vπ = kerTπ is the vertical bundle of π . Thus, we have:
1 1 1
(i) there exists a (unique) horizontal lift of every tangent vector
to M;
(ii) in fibred coordinates (xµ,yi,pµ) on J1π∗, then
i
∂ ∂
Vπ = span { , } ,H = span {H } ,
1 ∂yi ∂pµ µ
i
where H is the horizontal lift of ∂ .
µ ∂xµ
(iii) there is a horizontal projector h : TJ∗π −→ H.
3.3. Hamiltonian sections. Consider a hamiltonian section
h : J1π∗ −→ ΛnE
2
ofthecanonicalprojectionµ : ΛnE −→ J1π∗ whichinlocalcoordinates
2
read as
h(xµ,yi,pµ) = (xµ,yi,−H(x,y,p),pµ) .
i i
Denoteby Ω = h∗Ω , where Ω is the multisymplectic formonΛnE.
h 2 2 2
The field equations can be written as follows:
ihΩh = (n−1)Ωh , (3.1)
where h denotes the horizontal projection of an Ehresmann connection
in the fibred manifold π : J1π∗ −→ M.
1
The local expressions of Ω and Ω are:
2 h
Ω = −d(p dnx+pµdyi∧dn−1xµ)
2 0 i
Ω = −d(−H dnx+pµdyi∧dn−1xµ) .
h i
3.4. The field equations. Next, we go back to the Equation (3.1).
The horizontal subspaces are locally spanned by the local vector
fields
∂ ∂ ∂ ∂
H = h( ) = +Γi +(Γ )ν ,
µ ∂xµ ∂xµ µ ∂yi µ j ∂pν
j
where Γi and (Γ )ν are the Christoffel components of the connection.
µ µ j
Assume that τ is an integral section of h; this means that τ : M −→
J1π∗ is a local section of the canonical projection π : J1π∗ −→ M
1
such that Tτ(x)(T M) = H , for all x ∈ M.
x τ(x)
If τ(xµ) = (xµ,τi(x),τµ(x)) then the above conditions becomes
i
∂τi ∂H ∂τµ ∂H
= , i = −
∂xµ ∂pµ ∂xµ ∂yi
i
6 M. DE LEO´N,J.C. MARRERO,ANDD. MART´INDE DIEGO
which are the Hamilton equations.
4. The Hamilton-Jacobi theory
Let λ be a 2-semibasic n-form on E; in local coordinates we have
λ = λ (x,y)dnx+λµ(x,y)dyi∧dn−1xµ .
0 i
Alternatively, we can see it as a section λ : E −→ ΛnE, and then we
2
have
λ(xµ,yi) = (xµ,yi,λ (x,y),λµ(x,y)) .
0 i
A direct computation shows that
∂λ ∂λµ ∂λµ
dλ = 0 − i dyi∧dnx+ i dyj ∧dyi∧dn−1xµ .
(cid:18)∂yi ∂xµ(cid:19) ∂yj
Therefore, dλ = 0 if and only if
∂λ ∂λµ
0 = i (4.1)
∂yi ∂xµ
∂λµ ∂λµ
i = j . (4.2)
∂yj ∂yi
Using λ and h we construct an induced connection in the fibred
manifold π : E −→ M by defining its horizontal projector as follows:
h˜ : T E −→ T E
e e e
h˜e(X) = Tπ1,0 ◦h(µ◦λ)(e) ◦ǫ(X)
where ǫ(X) ∈ T (J1π∗) is an arbitrary tangent vector which
(µ◦λ)(e)
projects onto X.
From the above definition we immediately proves that
(i) h˜ is a well-defined connection in the fibration π : E −→ M.
(ii) The corresponding horizontal subspaces are locally spanned by
∂ ∂ ∂
H˜ = h˜( ) = +Γi((µ◦λ)(x,y)) .
µ ∂xµ ∂xµ µ ∂yi
The following theorem is the main result of this paper.
Theorem 4.1. Assume that λ is a closed 2-semibasic form on E and
that h˜ is a flat connection on π : E −→ M. Then the following condi-
tions are equivalent:
(i) If σ is an integral section of h˜ then µ ◦ λ ◦ σ is a solution of
the Hamilton equations.
(ii) The n-form h◦µ◦λ is closed.
HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 7
Before to begin with the proof, let us consider some preliminary
results.
We have
(h◦µ◦λ)(xµ,yi) = (xµ,yi,−H(xµ,yi,λµ(x,y)),λµ(x,y)) ,
i i
that is
h◦µ◦λ = −H(xµ,yi,λµ(x,y))dnx+λµdyi∧dn−1xµ .
i i
Notice that h◦µ◦λ is again a 2-semibasic n-form on E.
A direct computation shows that
∂H ∂H ∂λν ∂λµ
d(h◦µ◦λ) = − + j + i dyi∧dnx
(cid:18)∂yi ∂pν ∂yi ∂xµ(cid:19)
j
∂λµ
+ i dyj ∧dyi∧dn−1xµ .
∂yj
Therefore, we have the following result.
Lemma 4.2. Assume dλ = 0; then
d(h◦µ◦λ) = 0
if and only if
∂H ∂H ∂λν ∂λµ
+ j + i = 0 .
∂yi ∂pν ∂yi ∂xµ
j
Proof of the Theorem
(i) ⇒ (ii)
It should be remarked the meaning of (i).
Assume that
σ(xµ) = (xµ,σi(x))
is an integral section of h˜; then
∂σi ∂H
= .
∂xµ ∂pµ
i
(i) states that in the above conditions,
(µ◦λ◦σ)(xµ) = (xµ,σi(x),σ¯ν = λν(σ(x)))
j j
is a solution of the Hamilton equations, that is,
∂σ¯µ ∂λµ ∂λµ ∂σj ∂H
i = i + i = − .
∂xµ ∂xµ ∂yj ∂xµ ∂yi
8 M. DE LEO´N,J.C. MARRERO,ANDD. MART´INDE DIEGO
Assume (i). Then
∂H ∂H ∂λν ∂λµ
+ j + i
∂yi ∂pν ∂yi ∂xµ
j
∂H ∂H ∂λν ∂λµ
= + i + i , (since dλ = 0)
∂yi ∂pν ∂yj ∂xµ
j
∂H ∂σj ∂λν ∂λµ
= + i + i , (since the first Hamilton equation)
∂yi ∂xν ∂yj ∂xµ
= 0 (since (i))
which implies (ii) by Lemma 4.2.
(ii) ⇒ (i)
Assume that d(h◦µ◦λ) = 0.
Since h˜ is a flat connection, we may consider an integral section σ of
h˜. Suppose that
σ(xµ) = (xµ,σi(x)).
Then, we have that
∂σi ∂H
= .
∂xµ ∂pµ
i
Thus,
∂σ¯µ ∂λµ ∂λµ ∂σi
j = j + j ,
∂xµ ∂xµ ∂yi ∂xµ
∂λµ ∂λµ ∂σi
= j + i , (since dλ = 0)
∂xµ ∂yj ∂xµ
∂λµ ∂λµ ∂H
= j + i , (since the first Hamilton equation)
∂xµ ∂yj ∂pµ
i
∂H
= − , (since (ii)). (cid:3)
∂yj
Assume that λ = dS, where S is a 1-semibasic (n−1)-form, say
S = Sµdn−1xµ
Therefore, we have
∂Sµ ∂Sµ
λ = , λµ =
0 ∂xµ i ∂yi
and the Hamilton-Jacobi equation has the form
∂ ∂Sµ ∂Sµ
+H(xν,yi, ) = 0 .
∂yi (cid:18)∂xµ ∂yi (cid:19)
The above equations mean that
∂Sµ ∂Sµ
+H(xν,yi, ) = f(xµ)
∂xµ ∂yi
HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES 9
so that if we put H˜ = H − f we deduce the standard form of the
Hamilton-Jacobi equation (since H and H˜ give the same Hamilton
equations):
∂Sµ ∂Sµ
+H˜(xν,yi, ) = 0 .
∂xµ ∂yi
Analternative geometric approachof theHamilton-Jacobi theoryfor
Classical Field Theories in a multisymplectic setting was discussed in
[15, 16].
5. Time-dependent mechanics
A hamiltonian time-dependent mechanical system corresponds to a
classical field theory when the base is M = R.
We have the following identification Λ1E = T∗E and we have local
2
coordinates (t,yi,p ,p ) and (t,yi,p ) on T∗E and J1π∗, respectively.
0 i i
The hamiltonian section is given by
h(t,yi,p ) = (t,yi,−H(t,y,p),p ) ,
i i
and therefore we obtain
Ω = dH ∧dt−dp ∧dyi .
h i
If we denote by η = dt the different pull-backs of dt to the fibred
manifolds over M, we have the following result.
The pair (Ω ,dt) is a cosymplectic structure on E, that is, Ω and
h h
dt are closed forms and dt∧Ωn = dt∧Ω ∧···∧Ω is a volume form,
h h h
where dimE = 2n + 1. The Reeb vector field R of the structure
h
(Ω ,dt) satisfies
h
iR Ωh = 0 , iR dt = 1.
h h
The integral curves of R are just the solutions of the Hamilton equa-
h
tions for H.
The relation with the multisymplectic approach is the following:
h = R ⊗dt ,
h
or, equivalently,
∂
h( ) = R .
h
∂t
A closed 1-form λ on E is locally represented by
λ = λ dt+λ dyi.
0 i
Using λ we obtain a vector field on E:
(R ) = Tπ ◦R ◦µ◦λ
h λ 1,0 h
such that the induced connection is
h˜ = (R ) ⊗dt
h λ
10 M. DE LEO´N,J.C. MARRERO,ANDD. MART´INDE DIEGO
Therefore, we have the following result.
Theorem 5.1. The following conditions are equivalent:
(i) (R ) and R are (µ◦λ)-related.
h λ h
(ii) The 1-form h◦µ◦λ is closed.
Remark 5.2. An equivalent result to Theorem 5.1 was proved in [14]
(see Corollary 5 in [14]). ⋄
Now, if
∂S ∂S
λ = dS = dt+ dyi ,
∂t ∂yi
then we obtain the Hamilton-Jacobi equation
∂ ∂S ∂S
+H(t,yi, ) = 0 .
∂yi (cid:18) ∂t ∂yi (cid:19)
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