IJMMS 2004:57, 3045–3056
PII. S0161171204402282
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
HAMILTONIAN APPROACHES OF FIELD THEORY
CONSTANTIN UDRIŞTE and ANA-MARIA TELEMAN
Received 18 February 2004
We extend some results and concepts of single-time covariant Hamiltonian field theory to
the new context of multitime covariant Hamiltonian theory. In this sense, we point out the
role of the polysymplectic structure δ⊗J, we prove that the dual action is indefinite, we find
the eigenvalues and the eigenfunctions of the operator (δ⊗J)(∂/∂t) with periodic boundary
conditions, and we obtain interesting inequalities relating functionals created by the new
context. As an important example for physics and differential geometry, we study the multitime Yang-Mills-Witten Hamiltonian, extending the Legendre transformation in a suitable
way. Our original results are accompanied by well-known relations between Lagrangian and
Hamiltonian, and by geometrical explanations regarding the Yang-Mills-Witten Lagrangian.
2000 Mathematics Subject Classification: 53C07, 46E20, 55R10.
1. Introduction. As it is well known, the fields theories have a Lagrangian and Hamiltonian structure, since the interactions are mathematically modeled with energetic
Lagrangians-Hamiltonians (related by Legendre transformations) and their associated
action functionals.
The single-time Hamilton equations appeared for the first time in a paper of Lagrange
(1809) on perturbation theory, but it was Cauchy (1831) who first gave the true significance of those equations. In 1835, Hamilton put those equations at the basis of his
analytical mechanics and gave the first exact formulation of the least action principle.
The theory of single-time Hamilton equations with periodic boundary conditions was
initiated by Birkhoff (1917) who proved that the dual action is indefinite. Since then,
there were produced a lot of valuable papers that are now summarized in the book of
Mawhin and Willem [6].
The multitime Hamilton equations were first written in a book by de Donder [2].
But, to the authors’ knowledge, there is no prior work aimed at developing a theory
of multitime Hamilton equations with periodic boundary conditions. This is the first
subject of the present paper.
Some of the most important Lagrangians are those defined by Yang and Mills (1953)
and Witten [18] to produce modern explanations in quantum field theory (Maxwell or
Dirac equations). The second subject of the present paper is to introduce and study the
(single-time and multitime) Hamiltonian for a Yang-Mills-Witten functional.
Section 1 of this paper contains historical and bibliographical notes. Section 2 (Section 3) recalls the relations between the equations of single-time (multitime) first-order
Lagrangian field theory and the covariant Hamilton equations on the finite-dimensional symplectic (polysymplectic) phase space of covariant Hamiltonian field theory.
3046
C. UDRIŞTE AND A.-M. TELEMAN
Section 4 develops a theory of multitime Hamilton equations with boundary conditions.
Section 5 studies the Yang-Mills-Witten Hamiltonian.
2. Single-time Hamilton canonical equations. If L : R×R2n → R, (t, x i , ẋ i ) → L(t, x i ,
ẋ ) is a given Lagrangian, then the associated Euler-Lagrange equations are
i
∂L
d ∂L
=
,
dt ∂ ẋ i
∂x i
i = 1, . . . , n.
(2.1)
Hamilton (1834) simplified the structure of the Euler-Lagrange equations and turned
them into a form that has remarkable symmetry by
(1) introducing the conjugate momenta
pi =
∂L
;
∂ ẋ i
(2.2)
(2) considering the Hamiltonian
H = pi ẋ i − L
(2.3)
as a function of t, x, p.
We suppose that (2.2) defines, for every x, a diffeomorphism ẋ i → pi . This map is
called the Legendre transform.
Theorem 2.1. The Euler-Lagrange equations (2.1) are equivalent to the Hamilton
equations
dx i
∂H
,
=
dt
∂pi
dpi
∂H
=− i,
dt
∂x
i = 1, . . . , n.
(2.4)
Proof. The definitions (2.2) and (2.3) imply
∂H
∂L
∂L ∂ ẋ j
∂L
d
∂ ẋ j
= pj
−
−
= − i = − pi ,
i
i
i
∂x
∂x
∂x
∂ ẋ j ∂x i
∂x
dt
∂ ẋ j
∂H
∂L ∂ ẋ j
= ẋ i + pj
−
= ẋ i .
∂pi
∂pi ∂ ẋ j ∂pi
(2.5)
Consequently, the (second-order) ODEs (2.1) are equivalent to (first-order) ODEs (2.4).
2.1. Conservation of the Hamiltonian. Since
dpi i
dẋ i ∂L
d
d
∂L dx i
∂L dẋ i
∂L
H=
pi ẋ i − L =
ẋ + pi
−
−
−
=− ,
dt
dt
dt
dt
∂t ∂x i dt
∂ ẋ i dt
∂t
(2.6)
it follows that H is a first integral for the Hamilton equations if and only if the Lagrangian does not depend explicitly on t (autonomous Lagrangian).
HAMILTONIAN APPROACHES OF FIELD THEORY
3047
The Hamilton equations can be written in a compact form:
0
−δij
δij
0
 

∂H
dx j
 dt   ∂x i 
+
= 0 ,

 dp   ∂H 
0

∂pi
dt
where J =
δij
i
−δj 0
0
(2.7)
i
is the symplectic (complex) structure on R2n .
The minimization of the dual single-time action, the eigenvalues and the eigenfunctions of J(d/dt) with periodic boundary conditions, and existence of periodic solutions
of single-time Hamilton equations were discussed in [6].
3. Multitime Hamilton canonical equations. Let L : Rp ×Rn ×Rnp → R, (t α , x i , xαi ) →
L(t α , x i , xαi ), xαi = ∂x i /∂t α , be a given Lagrangian. The associated Euler-Lagrange equations are
∂L
∂ ∂L
=
,
∂t α ∂xαi
∂x i
i = 1, . . . , n, α = 1, . . . , p.
(3.1)
In [2], de Donder turned these equations into a form that has remarkable symmetry.
He used
(1) the conjugate momenta
pkα =
∂L
∂xαk
;
(3.2)
(2) the Hamiltonian
H = pkα xαk − L
(3.3)
as a function of t α , x i , piα .
Here it is, of course, required that (3.2) defines, for every x = (x i ), a continuously
differentiable bijection xαi → piα . This map is called the Legendre transform.
Theorem 3.1. The Euler-Lagrange equations (3.1) are equivalent to the Hamilton
equations
∂x i
∂H
=
,
∂t α
∂piα
∂piα
∂H
=− i,
∂t α
∂x
i = 1, . . . , n, α = 1, . . . , p
(3.4)
(summation over α).
Proof. The definitions (3.2) and (3.3) imply
k
∂piα
∂L
∂L ∂xαk
∂L
∂H
α ∂xα
=
p
−
−
=
−
=
−
,
k
∂x i
∂x i ∂x i ∂xαk ∂x i
∂x i
∂t α
k
k
∂L ∂xβ
∂H
β ∂xβ
i
=
x
+
p
−
= xαi .
α
k
∂piα
∂piα ∂xβk ∂piα
(3.5)
The Euler-Lagrange (second-order) PDEs (3.1) are equivalent to Hamilton (first-order)
PDEs (3.4).
3048
C. UDRIŞTE AND A.-M. TELEMAN
3.1. Conservation of the energy-momentum tensor field. The energy-momentum
tensor field is defined by
Tβα = xβi
∂L
∂xαi
− δα
β L.
(3.6)
We compute the divergence of this tensor field. We find
i ∂
∂ ∂L i
∂L ∂xγ
∂L
Tβα = α xβi piα − δα
+
xα +
β
i
α
α
i
α
∂t
∂t
∂t
∂x
∂xγ ∂t
∂xβi
i
∂piα
∂L
∂L i
∂L ∂xγ
−
−
x
−
∂t
∂t α ∂t β ∂x i β ∂xγi ∂t β
α
∂pi
∂L
∂L
∂L
= xβi
−
− β =− β.
∂t α ∂x i
∂t
∂t
=
(3.7)
p α + xβi
α i
Consequently, the energy-momentum tensor field is conserved if and only if the Lagrangian L does not depend explicitly on t α (autonomous Lagrangian).
4. Minimization of dual multitime action. The Hamilton equations can be written
in the form
β
i
δα
β δj
∂pi
∂H
+
= 0,
∂t α ∂x j
i
−δα
β δj
∂x j
∂H
+
= 0.
∂t α ∂p β
i
We will use also the polysymplectic structure
i
0
δα
β δj
δ⊗J =
,
δ = δα
α i
β ,
−δβ δj
0
0
J=
−δij
(4.1)
δij
0
(4.2)
discovered by the first author [13, 16, 17]. In this sense, we introduce the notations
X = Rn ,
X ∗ = dual of X,
δ = idX⊗Y ,
Y = Rp ,
0
δ⊗J =
−ϑ(δ)
Y ∗ = dual of Y ,
ϕ(δ)
,
0
(4.3)
where ϕ, ϑ are the canonical isomorphisms:
ϕ : End(X ⊗ Y ) → Hom X ∗ ⊗ End Y , X ∗
ϑ : End(X ⊗ Y ) → Hom X ⊗ Y ∗ , X ⊗ Y ∗ .
(4.4)
Then
δ ⊗ J ∈ Hom
X ⊗ Y ∗ ⊕ X ∗ ⊗ End Y , X ∗ ⊕ X ⊗ Y ∗
and we can give the Hamilton equations the following more compact form:




∂H
∂x
 ∂x 



 ∂t 

.
 = −
(δ ⊗ J) 
 ∂H 
 ∂p 
∂p
∂t
(4.5)
(4.6)
HAMILTONIAN APPROACHES OF FIELD THEORY
3049
According to the standard Riemannian metric on Rn+np , which is given by the matrix
δij 0 G = 0 δαβ δ , the associated Hamilton multitime action ψ is given by
ij
ψ(u) =
Ω
ᏸ t, u, u dv,
Ω = [0, T ]p ⊂ Rp ,
T > 0,
(4.7)
where u = (x, p), u = ∂u/∂t, dv = dt 1 ∧ · · · ∧ dt p , and
1 ∂piα i ∂x i α
ᏸ t, u, u =
x
−
p
+ H(t, x, p)
2 ∂t α
∂t α i
α
0
xi
δij
∂x j
1 ∂pj
+ H t, x(t), p(t) .
,
−
=
βα
piα
0
δ δij
2 ∂t α
∂t β
(4.8)
As it is well known, the Euler-Lagrange equations associated with the action ψ are
equivalent to (3.4). The action ψ is indefinite. This is easily shown by substituting the
function u with the sequence of functions
uk = xk , pk ,
α
(t) = ciα sin λk t β ,
xki (t) = c i cos λk t β ,
pk,i
uk (t) = cos λk t β (δ ⊗ I)c − sin λk t β (δ ⊗ J)c, β = fixed,
(4.9)
into the previous integral; we choose λk = 2kπ /T , k ∈ Z, c = (c i , ciα ) ∈ Rn+np , c = 1.
It follows that
uk (t) = 1,
α
(4.10)
∂pi
xj
δij
0
∂x i
β j
,
−
∀k ∈ Z.
γ = λk cj c
µγ
α
µ
pj
0
δ δij
∂t
∂t
Consequently,
kπ p β i
T ci c +
ψ uk =
T
t, uk (t) dv → −∞ or + ∞
Ω
(4.11)
according to k → −∞ or k → +∞. That is why the direct method of the calculus of
variations cannot be applied directly and more sophisticated approaches like minimax
methods, isoperimetric natural constraints, or dual least action principles have to be
used. In a forthcoming paper, we will concentrate on the case where the Hamiltonian
H(t, u) is convex in u, in which case the dual least action principle seems to provide the
best results in the simplest way. Here we discuss the eigenvalues and the eigenfunctions
of the operator (δ ⊗ J)(∂/∂t) with periodic boundary conditions:
∂u
= λu(t),
(δ ⊗ J)
∂t
u(0, . . . , 0) = u k1 T , . . . , kp T ,
(4.12)
where λ ∈ R, ∂u/∂t is the matrix of elements ∂u/∂t α , and kα ∈ Z. The PDEs system
(4.12) is equivalent to
∂x i
= −λpiα ,
∂t α
∂piα
= λx i .
∂t α
(4.13)
3050
C. UDRIŞTE AND A.-M. TELEMAN
We get particular solutions by taking
x i (t) = c i cos λiα t α ,
β
β
pi = ci sin µiα t α ,
(4.14)
β
where λiα , µiα , c i , ci are real numbers such that
T λiα
∈ Z,
2π
c i λiα = λciα ,
λiα = µiα ,
ciα µiα = λc i .
(4.15)
Supposing that c i ≠ 0, we get
λ2 = δαβ λiα λiβ ,
i = 1, . . . , n.
(4.16)
We have the following theorem.
Theorem 4.1. The system (4.12) has a nontrivial solution of the form (4.14) if and
only if the equation T 2 λ2 = 4π 2 δαβ kα kβ has solutions (k1 , . . . , kp ) ∈ Zp .
Note 4.2. The PDEs system (4.12) is associated with the Lagrangian
λ
∂x
j
=
δij x i x j + δαβ δij xαi xβ .
L1 x,
∂t
2
(4.17)
We could consider the Hamiltonian
H2 (x, p) =
1
ij
β
gij x i x j + Gαβ piα pj ,
2
(4.18)
ij
with real and constant coefficients gij , Gαβ . This generalizes the Hamiltonian associated
with the linear oscillator. The associated PDEs system will be linear and with constant
coefficients:
∂x i
ij
β
= Gαβ pj ,
∂t α
∂piα
= −gij x j .
∂t α
(4.19)
β
β
pi (t) = ci exp λα t α
(4.20)
When we look for solutions of the form
x i (t) = c i exp λα t α ,
β
with constants c i , ci , we obtain the algebraic equations
ij
β
λα ciα = −gij c j .
(4.21)
ij
Gαβ + g ij λα λβ ciα = 0.
(4.22)
λα c i = Gαβ cj ,
Supposing that det(gij ) ≠ 0, we get further
c j = −g ij ciα λα ,
Remark 4.3. Since the set of eigenvalues λ of the differential operator δ ⊗ J is unbounded (from below and from above), the quadratic form
∂u
1
(t), u(t) dv
(4.23)
(δ ⊗ J)
u →
2 Ω
∂t
HAMILTONIAN APPROACHES OF FIELD THEORY
3051
is indefinite on the space
1
HΩ
= u : Rp → Rn+np | u = absolutely continuous,
∂u
∈ L 2 Rp .
kα ∈ Z ⇒ u t 1 , . . . , t p = u t 1 + k1 T , . . . , t p + kp T ,
∂t
(4.24)
1
Theorem 4.4. If u ∈ HΩ
, then
2
u(t)2 dv ≤ T
4π 2
Ω
2
∂u
dv.
(t)
∂t
Ω
(4.25)
Proof. By assumption, the function u has a Fourier expansion
u(t) =
(k1 ,...,kp
k 1 t1 + · · · + k p tp
c k1 , . . . , kp exp 2π i
.
T
)∈Zp
(4.26)
If we denote k = (k1 , . . . , kp ), then k2 = k21 + · · · + k2p . The Perseval equality implies
∂u 2
4π 2 k2 c k 1 , . . . , k p 2
Tp
∂t dv =
2
T
Ω
k ,...,k
p
1
≥
2
4π
T2
k1 ,...,kp
2 4π 2
T p k1 , . . . , k p = 2
T
Ω
u(t)2 dv.
(4.27)
1
Theorem 4.5. Every u ∈ HΩ
satisfies the inequality
√
∂u
pT
∂u
dv.
, u(t) dv ≥ −
(t)
(δ ⊗ J)
∂t
2π Ω ∂t
Ω
Proof. We use the deviation
ũ(t) = u(t) −
1
Tp
(4.28)
Ω
u(t)dv.
(4.29)
The Cauchy-Schwarz inequality and inequality (4.25) imply
Ω
∂u
∂u
, u(t) dv =
, ũ(t) dv
(δ ⊗ J)
∂t
∂t
Ω
2
1/2 1/2
ũ(t)2 dv
(δ ⊗ J) ∂u (t) dv
≥−
∂t
Ω
Ω
2
2
1/2 1/2
∂ ũ
T
∂u
(δ ⊗ J)
dv
dv
≥−
(t)
(t)
2π
∂t
Ω
Ω ∂t
2
√
∂u
pT
(t)
≥−
dv.
2π Ω ∂t
(δ ⊗ J)
(4.30)
5. Witten-type Hamiltonians. The study of the Yang-Mills-Witten-type functionals
[1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15], which was initially related to a modern approach of
the Maxwell or Dirac equations, gave extremely important results for the development
of the differential geometry. This theory which is both experimentally and theoretically grounded, became very interesting and with many possible applications and open
3052
C. UDRIŞTE AND A.-M. TELEMAN
problems for research. In this section, we will apply the Hamiltonian formalism to a
functional of this type.
Let Spinc (4) = Spin(4) ×Z2 S 1 = [SU(2) × SU(2)] ×Z2 S 1 . There is a natural identification
Spinc (4) ≡
a+ , a− ∈ U (2) × U (2); det a+ = det a− ,
(5.1)
which induces the projections
± : Spinc (4) → U(2),
(5.2)
det : Spinc (4) → S 1
(5.3)
and also the morphism
defined by det(a+ , a− ) = det(a+ ) = det(a− ) for any (a+ , a− ) ∈ Spinc (4). Let (X, g) be
a Riemannian, compact, and oriented 4-manifold. Let (P c , γ) be a Spinc (4)-structure in
(X, g) and Σ± = P c ×± C 2 , its associated spinor-vector bundles.
We will consider as well the complex line bundle det Σ+ ≡ det Σ− ≡ P c ×det C and
also the space of the unitary connections in det Σ+ , denoted by Ꮽ(det Σ+ ). The space
Ꮽ(det Σ+ ) has a natural structure of affine space modeled over the vector space iA1 (X),
where A1 (X) is the space of the C ∞ − 1-forms defined on the manifold X. Let
W : Ꮽ det Σ+ × A0 Σ+ → R,
2
1 FA 2 + 1 ψ44 + 1
W (A, ψ) = ∇Â ψ +
L
16
4
4
(5.4)
s|ψ|2
(5.5)
X
be the Witten-type functional, where (A, ψ) ∈ Ꮿ = Ꮽ(det Σ+ ) × A0 (Σ+ ), |ψ|2 = ψψ̄, the
norms · are L2 -norms, and  is the connection naturally induced in Σ+ by A and by
the Levi-Civita connection of (X, g).
We assume that the restriction of the Riemannian structure g to a coordinate neighborhood is Euclidean and we make our computations only over this neighborhood. It
follows that the Levi-Civita connection associated to g has its coefficients, the curvature,
and its scalar curvature all equal to 0. Under such conditions, we apply the single-time
or multitime Hamiltonian formalism, and we will use the symbol sigma with indices for
summation.
If the connection A has its local associated form ia, then ∇Â ψ = dψ + (1/2)iaψ for
any ψ ∈ A0 (Σ+ ). Of course, ψ = (ψ1 , ψ2 ) = (ϕ1 + iη1 , ϕ2 + iη2 ) and a = aα dt α .
Lemma 5.1. The Lagrangian associated to W is given by the formula
L(A, ψ) =
3 2 1 2 j 2 j 2 j 2 j 2 j j
j
aα
+ η
ϕ
+ ϕα + ηα + ϕ ηα − ηj ϕα aα
4
α=0 j=1
3
2 1
1 aα,β − aβ,α +
+
32 α,β=0
4
2 j 2 j 2 + η
ϕ
j=1
for any ψ = (ϕ1 + iη1 , ϕ2 + iη2 ) ∈ A0 (Σ+ ), a = aα dt α .
2
(5.6)
3053
HAMILTONIAN APPROACHES OF FIELD THEORY
5.1. Single-time Hamilton equations. We can assume, under a suitable gauge transformation, that a0 = 0.
Lemma 5.2. The momenta associated to L are
pα =
∂L
1
= aα,0 ,
∂aα,0
8
qj =
∂L
j
j
∂ϕ0
= 2ϕ0 ,
∂L
rj =
j
∂η0
j
= 2η0 ,
(5.7)
with α = 1, 2, 3 and j = 1, 2.
Lemma 5.3. The Hamilton density associated to W is
H =4
3
ps
2
+
s=1
−
2
1 2
q + rj2
4 j=1 j
2 3 1 2 j 2 j 2 j 2 j 2 j j
j
ϕ
+ ϕs + ηs + ϕ ηs − ηj ϕs as
as
+ η
4
s=1 j=1
3
2
1
1 −
as,t − as,t at,s −
16 s,t=1
4
2
ϕ
j 2
+ η
(5.8)
2
j 2
.
j=1
Theorem 5.4. The Hamilton generalized equations, associated to W , are
ȧs = 8p s ,
ṗ s =
ϕ̇j =
1
qj ,
2
η̇j =
1
rj ,
2
2
3
j j
1
1 j
as |ψ|2 +
ϕ ηs − ηj ϕ s +
as,tt − at,st ,
2
8
t=1
j=1
q̇j = |ψ|2 ϕj +
ṙj = |ψ|2 ηj +
3 1
t=1
2
at
3 1 2 j
j
at ϕ + 2ϕtt − at,t ηj ,
2
t=1
2
j
ηj + 2ηtt + at,t ϕj ,
(5.9)
j = 1, 2; s = 1, 2, 3.
These equations follow from direct computations. Also, the first coordinate t 0 has
been privileged and considered as a time variable.
5.2. Multitime Hamilton equations. There are a lot of problems in which there is no
reason to prefer one variable to the other by choosing it as time. In such cases, we use
those field theories which involve many time variables (multitime or multiparameter).
In this section, we study the same functional, but instead of a single-time one, we will
consider it as a multitime functional, with no privileged variable t α . We use for our computations the appropriate Hamiltonian formalism which has been given in Section 3.
This formalism has been also studied and developed in [3, 4, 13, 16, 17]. The momenta
associated to L will be denoted by
p αβ =
∂L
,
∂aα,β
with α, β = 0, 1, 2, 3 and j = 1, 2.
qjα =
∂L
j
∂ϕα
,
rjα =
∂L
j
∂ηα
,
(5.10)
3054
C. UDRIŞTE AND A.-M. TELEMAN
Lemma 5.5. The momenta associated to L are
p αβ =
1
aα,β − aβ,α ,
8
j
qjα = 2ϕα − aα ηj ,
j
rjα = 2ηα + aα ϕj ,
(5.11)
with α, β = 0, 1, 2, 3 and j = 1, 2.
Remark 5.6. Since the first 16 relations of Lemma 5.5 (p αβ = · · · , for α, β = 0, 1, 2, 3)
are not independent, the general conditions of the Hamiltonian formalism are not fulfilled. It is yet possible to extend the Hamiltonian formalism so that such cases become
workable, in various ways. One way is the following. Suppose we have a Lagrangian
L(t, y, y ) such that the generalized momenta
piα =
∂L
(5.12)
∂yαi
are not independent. It is possible to find functions fαi (y, p) such that (5.12) be verified
when we write yαi = fαi (y, p). Suppose we made such a choice and define the formal
Hamiltonian
Hf (y, p) = fαi piα − L y, f (y, p)
(5.13)
by treating the variables piα as if they were functionally independent. Then we get
j
j
∂piα
∂Hf
∂fα α
∂L
∂L ∂fβ
∂L
=
p
−
−
=
−
=
−
,
∂y i
∂y i j ∂y i ∂y j ∂y i
∂y i
∂t α
β
j
j
(5.14)
∂fβ β
∂Hf
∂L ∂fβ
∂y i
i
i
.
α = fα +
α pj −
α = fα =
j
∂pi
∂pi
∂t α
∂y ∂pi
β
The equations thus obtained,
∂Hf
∂y i
=
,
∂t α
∂piα
∂piα
∂Hf
=−
,
∂t α
∂y i
(5.15)
will be named the generalized canonical equations. Note that Hf , as a function of the
variables piα , depends on the choice of the functions fαi , but it becomes a well-defined
function on the momenta piα by taking into consideration (5.12).
We apply this generalized Hamiltonian formalism to the functional (5.5) under the
same restrictions introduced at the beginning of Section 5. The corresponding Lagrangian has been written in Lemma 5.1. The relations p αβ = (1/8)(aα,β −aβ,α ) = −p βα show
that the momenta are not independent. In this case, we can choose the functions fαβ
such that
aα,β = fαβ (u, p) = 4p αβ + gαβ (u, p),
where gαβ = gβα .
(5.16)
3055
HAMILTONIAN APPROACHES OF FIELD THEORY
Lemma 5.7. The Hamilton density associated to W and to the choice (5.16) is
Hf = 2
2
3
3
3
αβ 2
1 α 2 α 2 +
p αβ gαβ +
p
qj + rj
4 α=0 j=1
α,β=0
α,β=0
2
3
1
1 j α
+
η qj − ϕj rjα aα −
2 α=0 j=1
4
2
2
j 2 j 2 + η )
.
ϕ
(5.17)
j=1
Theorem 5.8. The Hamilton generalized equations associated to Hf are
β
∂rj
∂t β
∂ϕj
1
∂aα
αβ
=
4p
+
g
,
= − qjα + aα ηj ,
αβ
∂t β
∂t α
2
∂p αβ
1 α
1 j α
∂ηj
j
,
r
η qj − ϕj rjα ,
=
−
a
ϕ
=
α
j
∂t α
2
∂t β
2
2
β
2 2 ∂qj
1
α
k
k
= aα rj +
+ η
ϕ
ϕj ,
∂t β
2
k=1
2
2 2 α
k
k
= −aα qj +
+ η
ϕ
ηj , j = 1, 2; α, β = 0, 1, 2, 3.
(5.18)
k=1
These equations follow from direct computations.
Remarks 5.9. (1) In the particular case gαβ = 0, the solutions of the previous equations are subject to the conditions ∂aα /∂t β = −∂aβ /∂t α .
(2) From Lemma 5.5, it follows that p αβ = −p βα . When we take into consideration
these relations, any choice of the functions f αβ gives the same Hamilton density
H =2
2
3
3 αβ 2 1 α 2 α 2 +
p
qj + rj
4
α=0 j=1
α,β=0
3
2
1
1 j α
+
η qj − ϕj rjα aα −
2 α=0 j=1
4
2
ϕ
j 2
+ η
(5.19)
2
j 2
.
j=1
Suitable computations give the formal Hamilton equations
∂ηj
1 α
rj − aα ϕj ,
=
α
∂t
2
2
β
∂q
k 2 k 2 1 j α
1
j
α
j α
= − η qj − ϕ rj ,
= aα rj +
+ η
ϕ
ϕj ,
2
∂t β
2
k=1
2
2 2 1
= − aα qjα +
ϕ k + ηk
ηj , j = 1, 2; α, β = 0, 1, 2, 3.
2
k=1
∂aα
= 4p αβ ,
∂t β
∂p αβ
∂t β
β
∂rj
∂t β
∂ϕj
1 α
qj + aα ηj ,
=
α
∂t
2
(5.20)
These are the equations from Theorem 5.8 in the particular case gαβ = 0.
(3) The first formula of Theorem 5.8 requires the integrability conditions
αβ
∂gαβ ∂gαγ
∂ 2 aα
∂p
∂p αγ
∂ 2 aα
−
=
4
−
−
= 0.
+
∂t β ∂t γ ∂t γ ∂t β
∂t γ
∂t β
∂t γ
∂t β
(5.21)
3056
C. UDRIŞTE AND A.-M. TELEMAN
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Constantin Udrişte: Department of Mathematics I, University Politehnica of Bucharest, Splaiul
Independenţei 313, Bucharest 060042, Romania
E-mail address: udriste@vectron.mathem.pub.ro
Ana-Maria Teleman: Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, Bucharest 70109, Romania
E-mail address: amteleman@geometry.math.unibuc.ro