On the gauge structure of the calculus of variations with constraints

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by  Danilo Bruno

ON THE GAUGE STRUCTURE OF THE CALCULUS OF VARIATIONS WITH
CONSTRAINTS
D. BRUNO, G. LURIA, E. PAGANI
Abstract. A gauge–invariant formulation of constrained variational calculus, based on theintro-
ductionofthebundleofaffinescalars overtheconfigurationmanifold,ispresented. Intheresulting
setup,the“Lagrangian”L isreplacedbyasectionofasuitableprincipalfibrebundleovertheveloc-
ity space. A geometric rephrasement of Pontryagin’s maximum principle, showing the equivalence
1
1 between a constrained variational problem in thestate space and a canonically associated free one
0 in a higheraffine bundle,is proved.
2
n Keywords: Constrained calculus of variations; Optimal control theory; Gauge theory; First varia-
a tion.
J
Mathematical Subject Classification 2010: 49J, 70Q05, 37J60, 70H03, 53B05, 53D10
6
2

]
h
p
Introduction
-
h
t Almost ten years ago, a mathematical setting for a formulation of Classical Mechanics, automat-
a
m ically embodyingits gauge invariance, hasbeen introduced[2,3]. Besides that, amorerecentpaper
[1] proposes a geometric revisitation of the calculus of variations in the presence of non–holonomic
[

constraints.
1
v The present work uses the arguments of [1] and the geometrical framework provided by [2, 3] to
5 analyze the underlying gauge structure of constrained variational calculus. As a by–product, the
4
resultingschemeallows toobtaintherelevantingredientsthatarecommonly usedinthevariational
0
5 context as canonical geometrical objects.
.
1 The topic will be developed within the family of differentiable curves alone, thus avoiding all the
0
issues coming from the possible presence of corners which are of poor significance from the gauge–
1
1 theoretical point of view. The extension to the piecewise differentiable case, surely interesting on
:
v its own, can however be easily pursued.
i
X Consider an abstract system B, subject to a set of differentiable and possibly non–holonomic
r constraints, and let A denote its admissible velocity space.
a
Then, define an action functional by integrating a suitable differentiable“cost function”on A
— called the Lagrangian — along the admissible evolutions of the system. In its essential features,
the variational problem we shall deal with is the one of characterizing, among all those evolutions
of B which fulfil the restrictions imposed by the assigned constraints and connect a fixed pair of
configurations, the extremals (if at all) of the given action functional. As we shall see, the whole
topic has very close links with optimal control theory.
For the sake of convenience, the first part of the paper shall be devoted as a reference tool
consisting of a brief review of a few basic aspects of jet–bundle geometry and non–holonomic
geometry as well as of those contents of [1, 2, 3] that will be involved in the subsequent discussion.
1
2 D.BRUNO, G.LURIA, E.PAGANI
This will include, among other things, a revisitation of the Lagrangian and Hamiltonian bundles.
All results will be stated without any proofs nor comments.
Afterwards, we shall develop an algorithm able to establish a canonical correspondence between
the input data of the problem, namely the constraints and the Lagrangian function, and an al-
ternative free variational problem over a distinguished overlying affine bundle C(A) → A. In the
resulting setup, under suitable hypotheses, the gauge–independent problem in C(A) is proved to
be equivalent to the actual constrained one. This clarifies the geometrical essence of Pontryagin’s
method, based on the introduction of the costates.
As the finalelement, the circumstances underwhichthe correspondence between the solutions of
the variational problem in A and those of its associated problem in C(A) is 1−1 are investigated.
1. Geometric setup
1.1. Preliminaries. Let V −→t R denotean (n+1)–dimensionalfibrebundle,henceforthcalled
n+1
the event space and referred to local fibred coordinates t,q1,...,qn.
Every section γ: R → V , locally described as qi = qi(t), will be interpreted as an evolution
n+1
of an abstract system B, parameterized in terms of the independent variable t. The first jet–
π
space j (V ) −→ V is an affine bundle over V , modelled on the vertical space V(V ).
1 n+1 n+1 n+1 n+1
By the very definition of jet–bundle, every section γ: R → V may be lifted to a section
n+1
j (γ): R → j (V ), simply by assigning to each t ∈ R the tangent vector to γ. The section
1 1 n+1
j (γ) will be called the jet–extension of γ on j (V ).
1 1 n+1
Bothspaces j (V ) and V(V ) maybeviewedassubmanifoldsofthetangentspace T(V )
1 n+1 n+1 n+1
according to the identifications
(1.1a) j (V )= {z ∈ T(V ) hz,dti = 1}
1 n+1 n+1
(1.1b) V(V )= {v ∈ T(V ) hv,dti = 0}
n+1 n+1
The terminology is borrowed from Classical Mechanics, where B is identified with a material
system, the manifold V with its configuration space–time, the projection t : V → R with
n+1 n+1
the absolute time function and the jet–space j (V ) with the velocity space of B.
1 n+1
The geometry of the manifold j (V ) will be regarded as known. The reader is referred to
1 n+1
[1, 4] for the notation, the terminology and a thorough analysis. Unless otherwise stated, given any
set of local coordinates on V , the corresponding local jet–coordinate system on j (V ) will be
n+1 1 n+1
denoted by t,q1,...,qn,q˙1,...,q˙n.
The dual of the vertical bundle, henceforth denoted by V∗(V ) −π→ V , is, in view of eq.(1.1b),
n+1 n+1
canonically isomorphic to the quotient of the cotangent space T∗(V ) by the equivalence relation
n+1
π(σ) =π(σ′)
′
σ ∼ σ ⇐⇒
( σ−σ′ ∝ dt π(σ)
Every local coordinate system t,qi in V induces fibred coordinates t,qi,pˆ in V∗(V ), with
n+1 i n+1
∂
∗
pˆ(σˆ) := σˆ, ∀ σˆ ∈ V (V )
i ∂qi n+1
* (cid:18) (cid:19)π(σˆ)+
ON THE GAUGE STRUCTURE OF THE CALCULUS OF VARIATIONS WITH CONSTRAINTS 3
and transformation laws
∂qk
(1.2) t¯= t+c , q¯i = q¯i(t,q1,...,qn) , p¯ˆ = pˆ
i k ∂q¯i
The annihilator of the tangent distribution to the totality of the jet–extensions of sections γ is
a subspace C j (V ) of T∗ j (V ) , called the contact bundle.
1 n+1 1 n+1
Alternatively,(cid:0) this last(cid:1)may b(cid:0)e seen as(cid:1) the pull–back of the space V∗(Vn+1) through the map
π
j (V ) −→ V . As such, C j (V ) is, at the same time, a vector bundle over j (V ) and
1 n+1 n+1 1 n+1 1 n+1
anaffinebundleoverV∗(V ). Themanifold C j (V ) willbereferredtocoordinatest,qi,q˙i,pˆ,
n+1 (cid:0) (cid:1) 1 n+1 i
related in an obvious way to those in j (V ) and in V∗(V ). Every σ ∈ C j (V ) will be
1 n+1 (cid:0) (cid:1) n+1 1 n+1
called a contact 1–form over j (V ).
1 n+1 (cid:0) (cid:1)
1.2. Non–holonomic constraints. Let A denote an embedded submanifold of j (V ), fibred
1 n+1
over V . The situation, summarized into the following commutative diagram
n+1
i
A −−−−→ j (V )
1 n+1
(1.3)
π π
 
Vn+1 Vn+1
y y
provides the natural setting for the study of non–holonomic constraints.
Themanifold A isreferredtolocal fibredcoordinates t,q1,...,qn,z1,...,zr withtransformation
laws
(1.4) t¯= t+c, q¯i = q¯i(t,q1,...,qn), z¯A = z¯A(t,q1,...,qn,z1,...,zr)
while the imbedding i: A→ j (V ) is locally expressed as
1 n+1
∂ ψ1 ··· ψn
(1.5a) q˙i = ψi(t,q1,...,qn,z1,...,zr) , i= 1,...,n , rank = r
∂(z1 ··· zr)
(cid:13) (cid:0) (cid:1)(cid:13)
(cid:13) (cid:13)
or, alternatively, may be implicit represented as (cid:13) (cid:13)
(cid:13) (cid:13)
∂ g1 ··· gn−r
(1.5b) gσ t,q1,...,qn,q˙1,...,q˙n = 0 , σ = 1,...,n−r , rank = n−r
∂(q˙1 ··· q˙n)
(cid:0) (cid:1)
(cid:13) (cid:13)
(cid:0) (cid:1) (cid:13) (cid:13)
In the following, we shall not distinguish between the manifold A an(cid:13)d its image i(A)(cid:13)⊂ j1(Vn+1).
In the presence of non–holonomic constraints, an evolution γ: R → V is called admissible
n+1
if and only if its first jet–extension is contained in A, namely if there exists a section γˆ: R → A
satisfying j (π · γˆ) = i · γˆ. Expressing any section γˆ in coordinates as qi = qi(t), zA = zA(t), the
1
admissibility requirement takes the explicit form
dqi
(1.6) = ψi(t,q1(t),...,qn(t),z1(t),...,zr(t))
dt
The concepts of vertical vector and contact 1–form are easily extended to the submanifold A: as
usual, the vertical bundle V(A) is the kernel of the push–forward π∗: T(A) → T(Vn+1) while
4 D.BRUNO, G.LURIA, E.PAGANI
the contact bundle C(A) is the pull–back on A of the bundle C(j (V )), as expressed by the
1 n+1
commutative diagram
C(A) −−−−→ C(j (V )) −−−−→ V∗(V )
1 n+1 n+1
(1.7)
 i  π 
A −−−−→ j1(Vn+1) −−−−→ Vn+1
y y y
The latter allows to regard the contact bundle C(A) as a fibre bundle over the space V∗(V ),
n+1
identical to the pull-back of V∗(V ) through the map A→ V .
n+1 n+1
1.3. Infinitesimal deformations of sections. (i) Quitegenerally, givenasection γ :R→ V ,
n+1
a (weak) deformation of γ is a 1–parameter family of sections γ , ξ ∈(−ε,ε) depending differen-
ξ
tiably on ξ and satisfying γ = γ.
0
In the presence of non–holonomic constraints, a deformation γ is called admissible if and only
ξ
if each section γ : R → V is admissible in the sense of §1.2. In a similar way, a deformation
ξ n+1
γˆ of an admissible section γˆ : R → A is called admissible if and only if all sections γˆ : R → A
ξ ξ
are admissible.
By definition, the admissible sections γ : R → V are in 1–1 correspondence with the admis-
n+1
sible sections γˆ :R→ A through the relations
(1.8) γ = π·γˆ , j (γ) = i·γˆ
1
Every admissible deformation of γ may therefore be expressed as γ = π ·γˆ , being γˆ : R → A
ξ ξ ξ
an admissible deformation of γˆ.
In coordinates, preserving the representation γˆ : qi = qi(t), zA = zA(t), the admissible defor-
mations of γˆ are described by equations of the form
(1.9) γˆ : qi =ϕi(ξ,t), zA = ζA(ξ,t)
ξ
subject to the conditions
(1.10a) ϕi(0,t) = qi(t), ζA(0,t) = zA(t)
∂ϕi
(1.10b) = ψi(t,ϕi(ξ,t),ζA(ξ,t))
∂t
For each t ∈R, the curve ξ → γˆ (t) is called the orbit of the deformation γˆ through the point
ξ ξ
γˆ(t). The vector field along γˆ tangent to the orbits at ξ = 0 is called the infinitesimal deformation
associated with γˆ .
ξ
Setting
∂ϕi ∂ζA
(1.11) Xi(t) := , ΓA(t) :=
∂ξ ∂ξ
(cid:18) (cid:19)ξ=0 (cid:18) (cid:19)ξ=0
the infinitesimal deformation tangent to γˆ is described by the vector field
ξ
∂ ∂
(1.12) Xˆ = Xi(t) + ΓA(t)
∂qi ∂zA
(cid:18) (cid:19)γˆ (cid:18) (cid:19)γˆ
while equation (1.10b) is reflected into the relation
dXi ∂ ∂ϕi ∂ψi ∂ψi
(1.13) = = Xk + ΓA
dt ∂t ∂ξ ∂qk ∂zA
(cid:12)ξ=0 (cid:18) (cid:19)γˆ (cid:18) (cid:19)γˆ
(cid:12)
(cid:12)
(cid:12)
ON THE GAUGE STRUCTURE OF THE CALCULUS OF VARIATIONS WITH CONSTRAINTS 5
commonly referred to as the variational equation.
The infinitesimal deformation tangent to the projection γ = π ·γˆ is similarly described by the
ξ ξ
field
∂ϕi ∂ ∂
(1.14) X = π∗Xˆ = = Xi(t)
∂ξ ∂qi ∂qi
(cid:18) (cid:19)ξ=0 (cid:18) (cid:19)γ (cid:18) (cid:19)γ
(ii) We now want to sketch out the construction of a suitable geometrical environment for the
description of such infinitesimal deformations. In order to do so, we denote by V(γ) −→t R the
vector bundle over R formed by the totality of vertical vectors along γ, and by A(γˆ) −→t R the
analogousbundleformedbythetotalityofvectorsalong γˆ annihilatingthe1–formdt. Bothbundles
are referred to fibred coordinates — the former to t,vi and the latter to t,vi,wA — according to
the prescriptions
∂
X ∈ V(γ) ⇐⇒ X = vi(X)
∂qi
(cid:18) (cid:19)γ(t(X))
∂ ∂
Xˆ ∈ A(γˆ) ⇐⇒ Xˆ = vi(Xˆ) + wA(Xˆ)
∂qi ∂zA
(cid:18) (cid:19)γˆ(t(Xˆ)) (cid:18) (cid:19)γˆ(t(Xˆ))
As proved in [1], the first jet–bundle j (V(γ)) is canonically isomorphic to the space of vectors
1
along the jet–extension j (γ) annihilating the 1–form dt.
1
In jet–coordinates, the identification is expressed by the relation
∂ ∂
Z ∈ j (V(γ)) ⇐⇒ Z = vi(Z) + v˙i(Z)
1 ∂qi ∂q˙i
(cid:18) (cid:19)j1(γ)(t(Z)) (cid:18) (cid:19)j1(γ)(t(Z))
i
Thepush–forwardoftheimbedding A −→ j (V ), restrictedtothesubspace A(γˆ)⊂ T(A), makes
1 n+1
the latter into a subbundle of j (V(γ)). This gives rise to a fibred morphism
1
A(γˆ) −−−i∗−→ j (V(γ))
1
(1.15a) π∗ π∗
 
V(γ) V(γ)
y y
expressed in coordinates as
∂ψi ∂ψi
(1.15b) v˙i = vk + wA
∂qk ∂zA
(cid:18) (cid:19)γˆ (cid:18) (cid:19)γˆ
All previous results are then summarized into the following
Proposition 1.1. Let γ : R → V and γˆ : R → A denote two admissible sections, related by
n+1
equation (1.8). Then:
i) the infinitesimal deformations of γ and those of γˆ are respectively expressed as sections
X :R → V(γ) and Xˆ : R→ A(γˆ);
ii) a section X : R → V(γ) represents an admissible infinitesimal deformation of γ if and
only if its first jet–extension factors through A(γˆ), i.e. if and only if there exists a section
Xˆ : R → A(γˆ) satisfying j1(X) = i∗Xˆ; conversely, a section Xˆ : R → A(γˆ) represents
6 D.BRUNO, G.LURIA, E.PAGANI
an admissible infinitesimal deformation of γˆ if and only if it projects into an admissible
infinitesimal deformation of γ, i.e. if and only if i∗Xˆ = j1(π∗Xˆ).
Fromastructuralviewpoint,Proposition1.1pointsouttheperfectlysymmetricrolesrespectively
played by diagram (1.3) in the study of the admissible evolutions and by diagram (1.15a) in the
studyoftheinfinitesimaldeformations, thusenforcingtheideathatthesecondcontextisessentially
a“linearized counterpart”of the former one.
1.4. The gauge setup.
1.4.1. The Lagrangian bundles. Given any system subject to (smooth) positional constraints, we
introduce a double fibration P −π→ V −→t R, where:
n+1
i) V −→t R is the configuration space–time of the system;
n+1
ii) P −π→ V is a principal fibre bundle with structural group (R,+).
n+1
As a consequence of the stated definition, each fibre P := π−1(x), x ∈ V is an affine
x n+1
1–space. The total space P is therefore a trivial bundle, diffeomorphic in a non–canonical way
to the Cartesian product V ×R, called the bundle of affine scalars over V .
n+1 n+1
The action of (R,+) on P results into a 1–parameter group of diffeomorphisms ψ : P → P,
ξ
conventionally expressed through the additive notation
(1.16) ψ (ν) := ν +ξ ∀ξ ∈ R, ν ∈ P
ξ
Every map u: P → R satisfying the requirement
u(ν +ξ) = u(ν)+ξ
is called a (global) trivialization of P. If u, u′ is any pair of trivializations, the difference u−u′
is then (the pull–back of) a function over V . Moreover, every section ς: V → P determines
n+1 n+1
a trivialization u ∈ F(P) and conversely, being the relation between ς and u expressed by the
ς ς
condition
(1.17) ν = ς(π(ν)) + u (ν) ∀ν ∈P
ς
Therefore, once a (global) trivialization u: P → R has been chosen, every section ς: V → P is
n+1
completely characterized by the knowledge of the function f = ς∗(u) ∈ F(V ).
n+1
The assignment of u allows to lift every local coordinate system t,q1,...,qn over V to a
n+1
corresponding fibred one t,q1,...,qn,u over P, being the most general transformation between
fibred coordinates of the form
t¯= t+c , q¯i = q¯i(t,q1,...,qn) , u¯ = u+f(t,q1....,qn)
Theactionofthegroup (R,+) onthemanifold P isexpressedinfibredcoordinatesbytherelations
t(ν +ξ) =t(ν) , qi(ν +ξ) = qi(ν) , u(ν +ξ) = u(ν)+ξ
As a result, the generator of the group action (1.16), usually referred to as the fundamental vector
field of P, is canonically identified with the field ∂ .
∂u
The (pull–back of the) absolute time function determines a fibration P −→t R whose associated
first jet–space is indicated by j (P,R) −π→ P and is referred to local jet–coordinates t,qi,u,q˙i,u˙
1
subject to transformation laws
(1.18a) t¯= t+c , q¯i = q¯i(t,q1,...,qn) , u¯ = u+f(t,q1....,qn)
ON THE GAUGE STRUCTURE OF THE CALCULUS OF VARIATIONS WITH CONSTRAINTS 7
∂q¯i ∂q¯i ∂f ∂f
(1.18b) q¯˙i = q˙k + , u¯˙ = u˙ + q˙k + := u˙ +f˙
∂qk ∂t ∂qk ∂t
The manifold j (P,R) is naturally embedded into the tangent space T(P) through the identifi-
1
cation
j (P,R) = {z ∈ T(P) hz,dti = 1}
1
expressed in local coordinate as
∂ ∂ ∂
(1.19) z ∈ j (P,R) ⇐⇒ z = +q˙i(z) +u˙i(z)
1 ∂t ∂qi ∂u
(cid:20) (cid:21)π(z)
In addition to the jet attributes, the space j (P,R) inherits from P two distinguished actions of
1
the group (R,+), related in a straightforward way to the identification (1.19).
The former is simply the push–forward of the action (1.16), restricted to the submanifold
j (P,R) ⊂ T(P). In jet–coordinates, a comparison with equation (1.19) provides the local repre-
1
sentation
∂ ∂ ∂
(1.20a) (ψ ) (z) = + q˙i(z) + u˙i(z)
ξ ∗ ∂t ∂qi ∂u
(cid:20) (cid:21)π(z)+ξ
expressed symbolically as
(1.20b) (ψ ) : (t,qi,u,q˙i,u˙i) −→ (t,qi,u+ξ,q˙i,u˙i)
ξ ∗
The quotient of j (P,R) by this action is a (2n+2)–dimensional manifold, denoted by L(V ).
1 n+1
Thequotientmap makes j (P,R) intoa principalfibrebundleover L(V ), with structuralgroup
1 n+1
(R,+). Furthermore, by equation (1.20b), L(V ) is an affine fibre bundle over V with local
n+1 n+1
coordinates t,qi,q˙i,u˙.
The latter action of (R,+) on j (P,R) follows from the invariant character of the field ∂ and
1 ∂u
is expressed in local coordinates by the addition
∂ ∂ ∂ ∂
(1.21a) φ (z) := z+ξ = +q˙i(z) + u˙i(z)+ξ
ξ ∂u ∂t ∂qi ∂u
(cid:18) (cid:19)π(z) (cid:20) (cid:21)π(z)
(cid:0) (cid:1)
summarized into the symbolic relation
(1.21b) φ : (t,qi,u,q˙i,u˙i) −→ (t,qi,u,q˙i,u˙i +ξ)
ξ
The quotient of j (P,R) by this action is once again a (2n+2)–dimensional manifold, denoted by
1
Lc(V ). As before, equation (1.21b) points out the nature of Lc(V ) as a fibre bundle over P
n+1 n+1
(as well as on V ), with coordinates t,qi,u,q˙i. The quotient map makes j (P,R) → Lc(V )
n+1 1 n+1
into a principal fibre bundle, with structural group (R,+) and group action (1.21a).
Eventually, the group actions (1.20a), (1.21a) do commute. Therefore, each of them may be
used to induce a group action on the quotient space generated by the other one. This makes
both L(V ) and Lc(V ) into principal fibre bundles over a common“double quotient”space,
n+1 n+1
canonically diffeomorphic to the velocity space j (V ).
1 n+1
The situation is summarized into the commutative diagram
j (P,R) −−−−→ Lc(V )
1 n+1
 
L(Vn+1) −−−−→ j1(Vn+1)
y y
8 D.BRUNO, G.LURIA, E.PAGANI
in which all arrows denote principal fibrations, with structural groups isomorphic to (R,+) and
group actions obtained in a straightforward way from equations (1.20b), (1.21b). The principal
fibre bundles L(V ) → j (V ) and Lc(V ) → j (V ) are respectively called the Lagrangian
n+1 1 n+1 n+1 1 n+1
and the co–Lagrangian bundle over j (V ).
1 n+1
The advantage of this framework is most appreciated by giving up the traditional approach,
based on the interpretation of the Lagrangian function L(t,qi,q˙i) as the representation of a
(gauge–dependent) scalar field over j (V ) and introducing instead the concept of Lagrangian
1 n+1
section, meant as a section
ℓ : j (V )→ L(V )
1 n+1 n+1
of the Lagrangian bundle.
For each choice of the trivialization u of P, the description of ℓ takes the local form
(1.22) u˙ = L(t,qi,q˙i)
and so it does still rely on the assignment of a function L(t,qi,q˙i) over j (V ). However, as
1 n+1
soon as the trivialization is changed into u¯ = u + f, the representation (1.22) undergoes the
transformation law
(1.23) u¯˙ = u˙ +f˙ = L(t,qi,q˙i)+f˙:= L′(t,qi,q˙i)
involving a different, gauge–equivalent, Lagrangian.
1.4.2. The non-holonomic Lagrangian bundles. In the presence of non–holonomic constraints, the
construction of the Lagrangian bundles may be easily adapted to the submanifold A, through a
straightforward pull–back process.
The situation is conveniently illustrated by means of a commutative diagram
j (P,R) - Lc(V )
1 n+1
(cid:8)* (cid:8)*
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
jA(P,R) - Lc(A)
1
(1.24)
? ?
L(V ) - j (V )
n+1 1 n+1
(cid:8)* (cid:8)*
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
?(cid:8) ?(cid:8)
-
L(A) A
where:
• L(A) andLc(A) arerespectivelythepull–backofL(V ) andLc(V ) onthesubmanifold
n+1 n+1
A → j (V );
1 n+1
ON THE GAUGE STRUCTURE OF THE CALCULUS OF VARIATIONS WITH CONSTRAINTS 9
• the space jA(P,R) may be alternatively seen as the pull–back of the jet–bundle
1
j (P,R) → L(V ) on the submanifold L(A) → L(V ) or as the pull–back of
1 n+1 n+1
j (P,R) → Lc(V ) on Lc(A) → Lc(V ).
1 n+1 n+1
The geometrical properties of the above–defined pull–back bundles are straightforwardly inherited
from their respective holonomic counterparts. In particular:
• Every choice of a trivialization u of P allows to lift any coordinate system of A to coordi-
nates t,qi,zA,u onLc(A), t,qi,zA,u˙ on L(A) and t,qi,u,zA,u˙ on jA(P,R). Theresulting
1
coordinate transformations are obtained bycompleting equations (1.4)with (the significant
part of) the system
∂f ∂f
(1.25) u¯ = u+f(t,q1,...,qn) , u¯˙ = u˙ + + ψk(t,qi,zA):= u˙ +f˙
∂t ∂qk
• The embeddings L(A) → L(V ), Lc(A) → Lc(V ) as well as jA(P,R) → j (P,R) are
n+1 n+1 1 1
all locally described by equation (1.5a).
• Both actions (1.20a), (1.21a) of the group (R,+) on j (P,R) preserve the submanifold
1
jA(P,R) thereby inducing two corresponding actions (ψ ) and φ on jA(P,R), expressed
1 ξ ∗ ξ 1
in coordinate as
(1.26a) (ψ ) : t,qi,u,zA,u˙ −→ t,qi,u+ξ,zA,u˙
ξ ∗
(1.26b) φ : (cid:0)t,qi,u,zA,u˙(cid:1) −→ (cid:0)t,qi,u,zA,u˙ +ξ(cid:1)
ξ
Acting in the same way a(cid:0)s before, it is(cid:1)easily seen(cid:0)that the manifol(cid:1)d jA(P,R) is a principal
1
fibrebundleoverL(A)undertheaction (ψ ) , as wellas aprincipalfibrebundleoverLc(A)
ξ ∗
under the action φ . Moreover, both L(A) and Lc(A) are principal fibre bundles over A
ξ
under the (induced) actions (ψ ) and φ respectively. Accordingly, all arrows in the front
ξ ∗ ξ
and rear faces of diagram (1.24) express principal fibrations, while those in the left and
right–hand faces are principal bundle homomorphisms.
Preservingtheterminology, theprincipalbundlesL(A)→ AandLc(A)→ Aarerespectivelycalled
the non–holonomic Lagrangian bundle and thenon–holonomic co–Lagrangian bundle over A, while
any section ℓ: A → L(A) is referred to as a non–holonomic Lagrangian section.
Once a trivialization u of P has been fixed, the description of ℓ takes the local form
(1.27) u˙ = L(t,qi,zA)
which undergoes the transformation law
∂f ∂f
(1.28) u¯˙ = u˙ +f˙ = L(t,qi,zA)+ + ψi := L′(t,qi,zA)
∂t ∂qi
under an arbitrary change u→ u+f(t,q1,...,qn).
1.4.3. The Hamiltonian bundles. Parallelling the discussion in §1.4.1, we shall now briefly go over
the construction of the Hamiltonian bundles on V . To this end, we focus on the fibration
n+1
P → V , and denote by π: j (P,V ) → P the associated first jet–space.
n+1 1 n+1
10 D.BRUNO, G.LURIA, E.PAGANI
Every fibred coordinate system t,qi,u on P induces local coordinates t,qi,u,p ,p on j (P,V ),
0 i 1 n+1
with transformation group
(1.29a) t¯= t+c, q¯i = q¯i(t,q1,...,qn), u¯ = u+f(t,q1,...,qn)
∂f ∂f ∂qk ∂f ∂qk
(1.29b) p¯ = p + + p + , p¯ = p +
0 0 ∂t k ∂qk ∂t i k ∂qk ∂q¯i
(cid:18) (cid:19) (cid:18) (cid:19)
The manifold j (P,V ) is naturally imbedded into the cotangent space T∗(P) through the
1 n+1
identification
∂
∗
j (P,V ) = η ∈ T (P) η, = 1
1 n+1
∂u
(cid:26) (cid:28) (cid:29) (cid:27)
expressed in local coordinate as
(1.30) η ∈j (P,V ) ⇐⇒ η = du − p (η)dt − p (η)dqi
1 n+1 0 i π(η)
Furthermore, the jet–bundle structure endows j ((cid:2)P,V ) with a contact bu(cid:3)ndle, locally generated
1 n+1
by the Liouville 1–form
(1.31) Θ˜ = du − p dt − p dqi
0 i
which is an intrinsically defined object, as ensured by equations (1.29a,b).
Exactly as in theLagrangian case, one can easily establish twodistinguishedactions of thegroup
(R,+) on j (P,V ), expressed locally as
1 n+1
(1.32a) (ψξ)∗(η) := (ψ−ξ)∗∗(η) = du−p0(η)dt−pi(η)dqi π(η)+ξ
(1.32b) φ (η) := η−ξ(dt) =(cid:2) du−(p (η)+ξ)dt−p(cid:3)(η)dqi
ξ π(η) 0 i π(η)
(cid:2) (cid:3)
In this connection, we point out that:
• The direct product of the actions (1.32a,b) makes j (P,V ) into a principal fibre bundle
1 n+1
over a (2n+1)–dimensional base space Π(V ), with coordinates t,qi,p , called the phase
n+1 i
space.
• In view of equations (1.2), (1.29a,b), the phase space Π(V ) is readily seen as an affine
n+1
bundle over V , modelled on V∗(V ).
n+1 n+1
• The quotient of j (P,V ) by the action (1.32a), denoted by H(V ), is an affine bundle
1 n+1 n+1
over V , modelled on the cotangent space T∗(V ) and called the Hamiltonian bundle.
n+1 n+1
• Anytrivializationu: P → Rallowstolifteverylocalcoordinatesystem t,q1,...,qn onV
n+1
toacorrespondingone t,q1,...,qn,p ,p ,...,p onH(V ),subjecttothetransformation
0 1 n n+1
law
∂f ∂f
(1.33) p¯ = p + , p¯ = p +
0 0 ∂t i i ∂qi
further to a change of u into u¯ = u+f(t,q1,...,qn).
• The quotient map makes j (P,V ) into a principal fibre bundle over H(V ), with
1 n+1 n+1
structural group (R,+) and fundamental vector ∂ .
∂u
• The canonical Liouville 1–form (1.31) endows j (P,V )→ H(V ) with a distinguished
1 n+1 n+1
connection, called the Liouville connection.