Legendre Transformation and Analytical Mechanics

Legendre Transformation and Analytical Mechanics:
a Geometric Approach ∗
Enrico Massa
Dipartimento di Matematica dell’Università di Genova
Via Dodecaneso, 35 - 16146 Genova (Italia)
E-mail: [email protected]
Enrico Pagani
Dipartimento di Matematica dell’Università di Trento
Via Sommarive, 14 - 38050 Povo di Trento (Italia)
E-mail: [email protected]
Stefano Vignolo
Dipartimento di Matematica dell’Università di Genova
Via Dodecaneso, 35 - 16146 Genova (Italia)
E-mail: [email protected]
Abstract
A revisitation of the Legendre transformation in the context of affine principal
bundles is presented. The argument, merged with the gauge–theoretical considerations developed in [1], provides a unified representation of Lagrangian
and Hamiltonian Mechanics, extending to arbitrary non-autonomous systems
the symplectic approach of W.M. Tulczyjew.
PACS: 03.20+1
1991 Mathematical subject classification: 70D10, 58F05, 53C57
Keywords: Lagrangian dynamics, Legendre transformation, Symplectic geometry.
1
Introduction
In recent papers [1, 2, 3, 4] a new geometrical framework for Analytical Mechanics has been developed. The formulation automatically embodies the invariance
of Lagrange’s equations under gauge transformations of the form L → L + f˙,
f˙ denoting the total time derivative of an arbitrary smooth function over the configuration manifold. Within this context we shall introduce an enhanced version
∗
Research partly supported by the National Group for Mathematical Physics (GNFMINDAM).
of the Legendre transformation, and discuss its significance in the representation
of the equations of motion. The argument extends to arbitrary dynamical systems
the symplectic framework originally developed by Tulczyjew in time–independent
Mechanics [5, 6, 7, 8, 9].
The mathematical foundations of the method are dealt with in Section 2. The
central point is the introduction of an involutory notion of duality between affine
principal bundles. On this basis, the stated “enhanced version” of the Legendre
transformation is established.
The subsequent analysis, in Section 3, shows that the Lagrangian and Hamiltonian bundles, i.e. the cornerstones of the the gauge–theoretical formulation of
Dynamics developed in [1], satisfy the duality criterion. A straightforward comparison with the results of Section 2 completes the construction, giving rise to a
canonical diffeomorphism between higher jet bundles, essentially equivalent to the
so called Tulczyjew triple T ∗ (T ∗ (M )) ↔ T (T ∗ (M )) ↔ T ∗ (T (M )). The dynamical
implications of the scheme are discussed.
2
2.1
Affine spaces and affine bundles
Algebraic preliminaries
Let Q and X respectively denote an (n + 1)–dimensional affine space and a free
vector on Q, or, what is the same, a constant vector field on Q .
The quotient of Q by the 1-parameter group of translations ϕξ : x → x + ξX
generated by X is then an n–dimensional affine space M , while the quotient map
π : Q → M is an affine surjection, making Q into a principal fiber bundle over
M , with structural group (<, +) and fundamental vector field X.
In what follows, we shall refer M to (global) affine coordinates x1 , . . . , xn , and Q
∂
to fibered affine coordinates x1 , . . . , xn , u , with hX, dui = 1 ⇔ X = ∂u
.
∗
Let Q denote the family of affine sections σ : M → Q. In coordinates, every
σ ∈ Q∗ admits a representation of the form
u = v + yi xi
(2.1)
The coefficients y1 , . . . , yn , v form a global coordinate system on Q∗ . The following
assertions are entirely straightforward:
• Q∗ is an (n + 1)–dimensional affine space, modelled on the vector space W ∗
formed by the totality of affine functions over M (known in the literature as the
extended dual of M );
• the 1-parameter group of translations ϕξ : Q → Q acts in an obvious way on Q∗ ,
sending each section σ : u = v + yi xi into the section ϕξ · σ : u = v + ξ + yi xi .
The generator of this action is a constant vector field Y on Q∗ , expressed in
∂
coordinates as Y = ∂v
. Viewed as an element of the modelling space W ∗ , the
vector Y coincides with the constant function f (x) ≡ 1 ∀ x ∈ M .
Let M ∗ denote the quotient of Q∗ by the action σ → ϕξ · σ described above.
A straightforward argument shows that the variables y1 , . . . , yn form an affine
2
coordinate system on M ∗ , and that the quotient map Q∗ → M ∗ is an affine
surjection, making Q∗ into a principal fiber bundle over M ∗ , with structural group
(<, +) and fundamental vector field Y .
For each σ ∈ Q∗ , let Fσ : Q → < denote the associated trivialization of the
bundle Q → M . The correspondence z, σ → Fσ (z) is then a function on Q × Q∗ ,
henceforth denoted by F (z, σ). In coordinates, eq. (2.1) provides the expression
Fσ = u − v(σ) − yi (σ) xi
mathematically equivalent to the representation
F (xi , u, yi , v) = u − v − xi yi
(2.2)
Eq. (2.2) establishes a bi-affine pairing between the spaces Q and Q∗ , invariant
under the 1-parameter group of translations
(z, σ) 7→ (ϕξ (z), ϕξ · σ)
(2.3)
In terms of this pairing, the condition for a point z ∈ Q to belong to an affine
section σ ∈ Q∗ , or, equivalently, for a section σ ∈ Q∗ to contain a point z ∈ Q, is
expressed by the relation F (z, σ) = 0. In this respect, precisely in the same way as
an element σ ∈ Q∗ is an affine section σ : M → Q, a point z ∈ Q may be viewed
as an affine section z : M ∗ → Q∗ , described in coordinates as
v = u(z) − xi (z) yi
(2.4)
and with image space z(M ∗ ) identical to the affine subspace of Q∗ formed by the
totality of hyperplanes containing z.
The previous arguments allow a simple characterization of the first jet spaces
j1 (Q, M ) and j1 (Q∗, M ∗ ) associated with the fibrations Q → M and Q∗ → M ∗ .
Recalling the interpretation of the bi-affine pairing (2.2) we have in fact
Theorem 2.1 Both manifolds j1 (Q, M ) and j1 (Q∗, M ∗ ) are diffeomorphic to the
submanifold S ⊂ Q × Q∗ described by the equation F (z, σ) = 0 .
Proof: by definition, every pair (z, σ) ∈ S, meant as a point z ∈ Q and a hyperplane σ : M → Q through z is an element of j1 (Q, M ), while, meant as a point
σ ∈ Q∗ and a hyperplane z : M ∗ → Q∗ containing σ, it belongs to j1 (Q∗, M ∗ ).
A straightforward argument (left to the reader) shows that both correspondences
S → j1 (Q, M ) and S → j1 (Q∗, M ∗ ) obtained in this way are in fact diffeomorphisms. In view of Theorem 2.1, the bundles j1 (Q, M ) and j1 (Q∗, M ∗ ) may be identified. Depending on the context, we shall refer them to global coordinates xi , u, yi
or xi , v, yi , related to each other by the correspondence
u − v − xi yi = 0
3
(2.5)
The content of Theorem 2.1 is completed by the following observations:
a) both manifolds j1 (Q, M ) and j1 (Q∗, M ∗ ) are principal fiber bundles, with structural group (<, +) , and group actions obtained as jet-extensions of the maps
ϕξ : Q → Q and ϕξ : Q∗ → Q∗ described above. With the identifications stated in
Theorem 2.1, the resulting bundle structures are unified into the principal fibration
S → M × M ∗ associated with the 1-parameter group of translations (2.3).
Both coordinate systems xi , u, yi and xi , v, yi are fibered over M × M ∗ , with u, v
playing the role of trivializations of the bundle S → M × M ∗ , and with the fun∂
∂
damental vector field expressed indifferently as X̂ = ∂u
= ∂v
.
∗
∗
b) by construction, each manifold j1 (Q, M ), j1 (Q , M ) carries a distinguished
contact 1-form, known as the Liouville 1-form. Once again, with the identification
stated in Theorem 2.1, both Liouville 1-forms are unified into a single geometrical
object, expressed in coordinates as
ϑ = du − yi dxi = dv + xi dyi
(2.6)
and playing the role of a connection 1-form over the bundle S → M × M ∗ .
The exterior 2-form
Ω := −dϑ = dyi ∧ dxi
(2.7)
identical, up to a sign, to the curvature of ϑ, endows the manifold M × M ∗ with
a canonical symplectic structure.
The previous discussion, summarized into the diagram
j1 (Q, M )
?
?
M
?
Q∗
M × M∗
Q
?
j1 (Q∗, M ∗ )
S
(2.8)
HH
HH
?
H
j
H
M∗
is the core of the Legendre transformation between affine bundles.
Every section ϕ : M → Q may in fact be lifted to a map j1 (ϕ) : M → j1 (Q, M ) ,
thereby giving rise, through the diagram (2.8), to correspondences Λϕ : M → Q∗ ,
λϕ : M → M ∗ and κϕ : M → M × M ∗ .
The last one is nothing but the graph κϕ (x) = (x, λϕ (x)) of the map λϕ . As far
as the other two are concerned, expressing ϕ as u = ϕ(x1 , . . . , xn ) , and recalling
eq. (2.5), as well as the definition of j1 (ϕ) , we get the coordinate representations
Λϕ :
λϕ :
∂ϕ
∂xi
∂ϕ
yi =
∂xi
; v = ϕ − xi
yi =
∂ϕ
∂xi
(2.9a)
(2.9b)
4
In view of eqs. (2.7), (2.9b), the map κϕ satisfies the identity
∂ϕ
∗
∗
i
i
κϕ (Ω) = λϕ (dyi ) ∧ dx = d
dx = 0
∂xi
(2.10)
indicating that the graph of λϕ is a Lagrangian submanifold of M × M ∗ .
In a perfectly symmetric way, every section ψ : M ∗ → Q∗ , lifted to a map
j1 (ψ) : M ∗ → j1 (Q∗, M ∗ ), gives rise, through the diagram (2.8), to correspondences
Λψ : M ∗ → Q , λψ : M ∗ → M and κψ : M ∗ → M × M ∗ , with κψ representing
the graph of λψ . In coordinates, expressing ψ as v = −ψ(y1 , . . . , yn ), we have
the representations
Λψ :
xi =
∂ψ
∂yi
λψ :
xi =
∂ψ
∂yi
; u = −ψ + yi
∂ψ
∂yi
(2.11a)
(2.11b)
Once again, in view of eq. (2.11b), the graph of λψ is easily recognized to be a
Lagrangian submanifold of M × M ∗ .
A special instance of the previous construction occurs when the map λϕ associated with the section ϕ : M → Q is a diffeomorphism, i.e. when the graph
κϕ (M ) ⊂ M × M ∗ projects injectively onto M ∗ . Under the stated assumption,
∗
∗
∗
∗
the correspondence ψ := Λϕ · λ−1
ϕ : M → Q is a section of the bundle Q → M ,
described in coordinates as
v = −yi xi + ϕ x1 , . . . , xn := −ψ (y1 , . . . , yn )
(2.12)
with the variables xi defined implicitly in terms of the yi ’s through eqs. (2.9b).
From eqs. (2.9b), (2.12), by elementary computations, we get the identities
∂ψ
∂xj
∂ϕ ∂xj
= xi − yj
− j
= xi
∂yi
∂yi
∂x ∂yi
;
ϕ = −ψ + yi
∂ψ
∂yi
(2.13)
Comparison with eqs. (2.11a, b) provides the identifications
λψ = λ−1
ϕ
ϕ = Λψ · λ ϕ
;
pointing out the perfectly symmetric role played by the sections ϕ and ψ.
Consistently with the current terminology, every diffeomorphism M ←→ M ∗
arising from a section ϕ : M → Q through the algorithm indicated above will be
called a Legendre transformation.
2.2
Affine principal fibrations
The construction of Subsection 2.1 is easily extended to the context of affine bundles over an arbitrary base manifold N . The basic structure is summarized into
5
the diagram
Q −−−−→


πy
N
M

π
y
(2.14)
N
in which Q → N and M → N are affine fibrations, while Q → M is both an
affine bundle homomorphism, fibered over N , and a principal fiber bundle, with
structural group (<, +) 1 .
We shall refer the manifold N to local coordinates ξ 1 , . . . , ξ r . The bundles M and
Q will be respectively referred to affine fibered coordinates ξ 1 , . . . , ξ r , x1 , . . . , xn
and ξ 1 , . . . , ξ r , x1 , . . . , xn , u. The fibers of M → N and Q → N will be denoted
by Mξ , Qξ , ξ ∈ N .
For each ξ ∈ N let us now consider the family of affine sections σξ : Mξ → Qξ .
As pointed out in Subsection 2.1, these form an affine space Qξ ∗ carrying an affine
principal fibration Qξ ∗ → Mξ ∗ “dual” of the fibration Qξ → Mξ in the sense
described by eq. (2.2).
S
S
Introducing the spaces Q∗ := ξ∈N Qξ ∗ , M ∗ := ξ∈N Mξ ∗ , the situation is summarized into the commutative diagram
Q∗ −−−−→


πy
N
M∗

π
y
(2.15)
N
in which all vertical arrows represent affine fibrations, while Q∗ → M ∗ is an affine
bundle homomorphism and a principal fibration.
Exactly as in Subsection 2.1, every coordinate system ξ α , xi , u on Q determines
coordinates ξ α , yi , v on Q∗ and ξ α , yi on M ∗ on the basis of the requirement
ξ α (σ) := ξ α (π(σ)) ,
v(σ) + yi (σ) xi (x) = u (σ(x))
∀ x ∈ Mπ(σ)
(2.16)
Once again, the fibrations Q → M and Q∗ → M ∗ are dual of each other under
the bi-affine pairing (z, σ) → F (z, σ) defined on the fibered product Q ×N Q∗ by
F (z, σ) = u(z) − v(σ) − yi (σ) xi (z)
(2.17)
In particular, denoting by S the submanifold of Q×N Q∗ described by the equation
F (z, σ) = 0 and recalling the proof of Theorem 2.1, we get the identifications
[
[
(2.18)
S=
j1 (Qξ , Mξ ) =
j1 (Qξ ∗, Mξ ∗ )
ξ∈N
ξ∈N
1
Basically, this means that M is the quotient of Q by the action of the 1-parameter group
of affine translations generated by an everywhere non-zero vector field X tangent to the fibers of
Q → N , and constant along each fiber.
6
Depending on the context, we shall refer S to coordinates ξ α, xi, u, yi or ξ α, xi, yi , v,
with transformation law
u − v − xi yi = 0
(2.19)
The previous arguments help analyzing the relationship between the first jet
spaces j1 (Q, M ) and j1 (Q∗, M ∗ ) . To this end, prior to any further consideration,
we recall that both spaces carry natural actions of the group (<, +), respectively
obtained by lifting the group actions on Q → M and on Q∗ → M ∗ . Introducing
the notation B := j1 (Q, M )/(<, +) , B ∗ := j1 (Q∗, M ∗ )/(<, +), the situation is
expressed diagrammatically as
j1 (Q, M ) −−−−→


y
B
j1 (Q∗, M ∗ ) −−−−→


y
Q


y
B∗
−−−−→ M
Q∗


y
(2.20)
−−−−→ M ∗
the vertical arrows denoting affine principal fibrations.
Using jet coordinates ξ α, xi , u, uα , ui on j1 (Q, M ) and ξ α, yi , v, vα , v i on j1 (Q∗, M ∗ ),
the fundamental vector fields over j1 (Q, M ) → B and j1 (Q∗, M ∗ ) → B ∗ coincide
∂
∂
respectively with the fields ∂u
and ∂v
.
In addition to this, the manifolds j1 (Q, M ), j1 (Q∗, M ∗ ) are endowed with corresponding Liouville 1-forms, expressed in coordinates as
Θ1 = du − uα dξ α − ui dxi
(2.21a)
Θ2 = dv − vα dξ α − v i dyi
(2.21b)
and playing the role of connection 1-forms with respect to the principal fibrations
j1 (Q, M ) → B and j1 (Q∗, M ∗ ) → B ∗ discussed above.
Finally, by definition, for each z ∈ Q, the elements of the fiber j1 (Q, M )|z
are equivalence classes of sections ϕ : M → Q having a first order contact at z.
Setting ξ := π(z) ∈ N , the restriction of each such ϕ to the fiber Mξ is a section
ϕξ : Mξ → Qξ . Moreover, if two sections ϕ, ϕ0 have a first order contact at z, the
restrictions ϕξ , ϕ0ξ also do.
S
In this way, by varying z, we obtain a surjection j1 (Q, M ) → ξ∈N j1 (Qξ , Mξ ) .
S
A similar argument establishes the surjection j1 (Q∗, M ∗ ) → ξ∈N j1 (Qξ ∗ , Mξ ∗ ) .
In view of eq. (2.18) this makes both j1 (Q, M ) and j1 (Q∗, M ∗ ) into fiber bundles
over the same base manifold S. On this basis, we state
Theorem 2.2 There exists a unique diffeomorphism ψ : j1 (Q, M ) → j1 (Q∗, M ∗ )
making the diagram
ψ
j1 (Q, M ) −−−−→ j1 (Q∗, M ∗ )




y
y
S
S
commutative, and satisfying ψ ∗ (Θ2 ) = Θ1 .
7
(2.22)
Proof: in coordinates, on account eqs. (2.18), (2.19), the requirement of commutativity of the diagram (2.22) is expressed by the relations
ψ ∗ (ξ α ) = ξ α ,
ψ ∗ (yi ) = ui ,
ψ ∗ (v i ) = −xi ,
ψ ∗ (v) = u − xi ψ ∗ (yi ) = u − xi ui
Comparison with eqs. (2.21a, b) provides the evaluation
ψ ∗ (Θ2 ) = d(u − xi ui ) − ψ ∗ (vα ) dξ α + xi dui = Θ1 + [uα − ψ ∗ (vα )] dξ α
showing that the condition ψ ∗ (Θ2 ) = Θ1 requires the further identification
ψ ∗ (vα ) = uα
This establishes at one time the existence and the uniqueness of a diffeomorphism
ψ : j1 (Q, M ) → j1 (Q∗, M ∗ ) satisfying all stated requirements. In view of Theorem 2.2, the bundles j1 (Q, M ) → B and j1 (Q∗, M ∗ ) → B ∗ may be
regarded as different copies of the same abstract bundle, henceforth denoted by
J → B . Depending on the context, we shall refer the latter to fibered coordinates
ξ α, xi , ηα , yi , u or ξ α, xi , ηα , yi , v, related to each other by the transformation law
u − v − xi yi = 0
(2.23a)
and to the ordinary jet-coordinates ξ α, xi , u, uα , ui on j1 (Q, M ) and ξ α, yi , v, vα , v i
on j1 (Q∗, M ∗ ) by the further identifications
u i = yi ,
v i = −xi ,
uα = vα = ηα
(2.23b)
∂
∂
As a result, the fundamental vector fields ∂u
and ∂v
get identified. In a similar
way, the Liouville 1-forms (2.21a, b) collapse into a single geometrical object,
henceforth denoted by Θ, expressed in coordinates as
Θ = du − ηα dξ α − yi dxi = dv − ηα dξ α + xi dyi
(2.24)
Ω := −dΘ = dηα ∧ dξ α + dyi ∧ dxi
(2.25)
The 2-form
endows the base manifold B with a canonical symplectic structure.
The previous arguments, summarized into the diagram
j1 (Q∗, M ∗ )
J
j1 (Q, M )
?
?
Q
?
Q∗
B
H
?
M
(2.26)
HH
?
HH
j M∗
allow the construction of a completely involutory Legendre transformation between
8
the bundles Q → M and Q∗ → M ∗ . The line of approach, similar to the one
exploited in Section 2.1, may be traced as follows: every section ϕ : M → Q may
be lifted to a map j1 (ϕ) : M → j1 (Q, M ) , thereby giving rise, through the diagram
(2.26), to correspondences Λϕ : M → Q∗ , λϕ : M → M ∗ and κϕ : M → B.
In coordinates, expressing ϕ as u = ϕ(ξ 1 , . . . , ξ r, x1 , . . . , xn ) , and recalling
eq. (2.23a), as well as the definition of j1 (ϕ) , we get the representations
Λϕ :
λϕ :
κϕ :
∂ϕ
∂xi
∂ϕ
yi =
∂xi
∂ϕ
yi =
∂xi
; v = ϕ − xi
yi =
∂ϕ
∂xi
(2.27a)
(2.27b)
; ηα =
∂ϕ
∂ξ α
(2.27c)
In view of eqs. (2.25), (2.27c), the map κϕ satisfies the identity
∂ϕ
∂ϕ
∗
∗
α
i
α
i
κϕ (Ω) = κϕ dηα ∧ dξ + dyi ∧ dx = d
dξ + i dx = 0
∂ξ α
∂x
(2.28)
showing that the image κϕ (M ) is a Lagrangian submanifold of B.
In a perfectly symmetric way, every section ψ : M ∗ → Q∗ , lifted to a map
j1 (ψ) : M ∗ → j1 (Q∗, M ∗ ), induces correspondences Λψ : M ∗ → Q , λψ : M ∗ → M
and κψ : M ∗ → B. The implementation in coordinates is entirely straightforward,
and is left to the reader.
Finally, when the map λϕ associated with the section ϕ : M → Q is a diffeo∗
∗
morphism, the correspondence ψ := Λϕ · λ−1
ϕ : M → Q is a section of the bundle
∗
∗
Q → M , described in coordinates as
v = −yi xi + ϕ ξ 1 , . . . , ξ r, x1 , . . . , xn := −ψ ξ 1 , . . . , ξ r, y1 , . . . , yn
(2.29)
with the functions xi ξ 1 , . . . , ξ r, y1 , . . . , yn defined implicitly by eqs. (2.27b).
From eqs. (2.27b), (2.29), by elementary computations, we get the identities
∂ψ
∂xj
∂ϕ ∂xj
= xi − yj
− j
= xi
∂yi
∂yi
∂x ∂yi
;
ϕ = −ψ + yi
∂ψ
∂yi
(2.30)
Comparison with eqs. (2.27a, b) provides the identifications
λψ = λ−1
ϕ
ϕ = Λψ · λ ϕ
;
pointing out once again the symmetric role played by the sections ϕ and ψ
The previous arguments extend to jet-bundles the classical approach to the
Legendre transformation developed by Tulczyjew [5, 8]. In this connection, see
also [12].
9
3
Classical Mechanics
3.1
Lagrangian and Hamiltonian bundles
A well known feature of Classical Mechanics is the invariance of Lagrange’s equations under gauge transformations of the form L → L + f˙ involving the total time
derivative of an arbitrary smooth function over the configuration manifold. This
fact is conveniently accounted for by working in an environment in which gauge
equivalent Lagrangians may be thought of as different representations of the same
geometrical object. The geometrical set-up, worked out in detail in [1, 2], relies
t
π
→ <, in which
on the introduction of a double fibration P −
→ Vn+1 −
• Vn+1 is the configuration space-time of the dynamical system in study, with
t
→ < representing absolute time;
the fibration Vn+1 −
π
• P −
→ Vn+1 is a principal fiber bundle, with structural group (<, +), called
the bundle of affine scalars over Vn+1 .
In what follows, we shall refer the manifold Vn+1 to local coordinates t, q i , and
P to fibered local coordinates t, q i , u (i = 1 . . . , n) , u denoting any trivialization
π
of P → Vn+1 . The first jet bundles associated with the fibration P −
→ Vn+1 and
t·π
with the composite fibration P −−→ < , respectively denoted by j1 (P, Vn+1 ) and
j1 (P, <) , will be referred to jet-coordinates t, q i , u, p0 , pi and t, q i , u, q̇ i , u̇ .
The manifold j1 (P, <) provides the basic environment for the gauge-invariant
Lagrangian formulation of Mechanics. As illustrated in [1, 2], the latter carries two
mutually commuting actions of the group (<, +), locally generated by the vector
∂
fields ∂u
and ∂∂u̇ , and giving rise to corresponding quotient spaces and quotient
maps. The situation is summarized into the diagram
j1 (P, <) −−−−→ Lc (Vn+1 )




y
y
(3.1)
L(Vn+1 ) −−−−→ j1 (Vn+1 )
in which all arrows express principal fibrations with structural group (<, +), while
j1 (Vn+1 ) := j1 (Vn+1 , <) denotes the velocity space of the system.
More specifically, the manifold L(Vn+1 ), with coordinates t, q i , q̇ i , u̇, is the quo∂
tient of j1 (P, <) by the action generated by ∂u
. The 1-parameter group generated
∂
by ∂ u̇ makes L(Vn+1 ) into a principal fiber bundle over j1 (Vn+1 ), known as the
Lagrangian bundle. Every section l : j1 (Vn+1 ) → L(Vn+1 ), expressed locally as
u̇ = L(t, q i , q̇ i ) , is called a Lagrangian section.
In a similar way, the quotient of j1 (P, <) by the action generated by ∂∂u̇ is
denoted by Lc (Vn+1 ). The principal fiber bundle Lc (Vn+1 ) → j1 (Vn+1 ), with
∂
structural group generated by ∂u
, is called the co-Lagrangian bundle.
10
As pointed out in [1], the use of Lagrangian sections in place of Lagrangian
functions automatically embodies the gauge invariance of the theory under arbitrary transformations L → L + f˙, and establishes a natural interpretation of the
Poincaré–Cartan 1-form as a connection 1-form over the co-Lagrangian bundle.
The Hamiltonian counterpart of the construction stems from an analysis of the
fibration P → Vn+1 . Once again, the first-jet space j1 (P, Vn+1 ) is endowed with
two mutually commuting actions of the group (<, +), now generated by the vector
∂
fields ∂u
, ∂p∂ 0 . These give rise to corresponding quotient spaces and quotient
maps, summarized into the diagram
j1 (P, Vn+1 ) −−−−→ Hc (Vn+1 )




y
y
H(Vn+1 )
(3.2)
−−−−→ Π(Vn+1 )
in which all arrows express principal fibrations with structural group (<, +). The
double quotient Π(Vn+1 ) is called the phase space of the system.
More specifically: the manifold H(Vn+1 ), with coordinates t, q i , p0 pi , is the
∂
quotient of j1 (P, Vn+1 ) by the action generated by ∂u
. The action generated
∂
by ∂p0 makes H(Vn+1 ) into a principal fiber bundle over Π(Vn+1 ), known as the
Hamiltonian bundle. Every section h : Π(Vn+1 ) → H(Vn+1 ), described in coordinates as p0 = H(t, q i , pi ) is called a Hamiltonian section.
In a similar way, the quotient of j1 (P, Vn+1 ) by the action generated by ∂p∂ 0 is
denoted by Hc (Vn+1 ). The principal fiber bundle Hc (Vn+1 ) → Π(Vn+1 ), with
∂
structural group generated by ∂u
, is called the co-Hamiltonian bundle.
The geometrical environment described by diagram (3.2) provides the starting
point for a gauge-invariant formulation of Hamiltonian Mechanics. A thorough
analysis of this point may be found in [1, 2, 3] and references therein. For the
present purposes we simply remind that the Liouville 1-form of j1 (P, Vn+1 ), expressed in coordinates as
ϑ := du − p0 dt − pi dq i
(3.3)
determines a connection over the principal fiber bundle j1 (P, Vn+1 ) → H(Vn+1 ).
The curvature of ϑ, described, up to a sign, by the exterior 2-form
Ω := −dϑ = dp0 ∧ dt + dpi ∧ dq i
(3.4)
endows the base manifold H(Vn+1 ) with a canonical symplectic structure.
3.2
Higher jet spaces
The algorithm developed in Subsection 2.2 applies in a natural way to the Lagrangian and Hamiltonian bundles described in Subsection 3.1, thereby providing a
mathematical environment for a unified formulation of time-dependent Lagrangian
and Hamiltonian Dynamics.
11
To start with, let us focus on the commutative diagram
L(Vn+1 ) −−−−→ j1 (Vn+1 )




y
y
Vn+1
(3.5)
Vn+1
and observe that, by construction, both j1 (Vn+1 ) → Vn+1 and L(Vn+1 ) → Vn+1
are affine bundles (the second one identical to the quotient of j1 (P, <) → P by the
∂
action generated by the vector field ∂u
), while the map L(Vn+1 ) → j1 (Vn+1 ) is at
the same time an affine bundle homomorphism, fibered over Vn+1 , and a principal
fiber bundle, with structural group (<, +).
The diagram (3.5) is therefore an example of affine principal fibration in the sense
described in Subsection 2.2.
In a perfectly symmetric way, the diagram
H(Vn+1 ) −−−−→ Π(Vn+1 )




y
y
Vn+1
(3.6)
Vn+1
defines another affine principal fibration over the same base space Vn+1 .
More specifically, from the discussion of Subsection 3.1 we can draw the following
conclusions:
• the bundle H(Vn+1 ) → Vn+1 is canonically isomorphic to the space of connections over the principal fiber bundle P → Vn+1 . At each x ∈ Vn+1 , the
elements of the fiber H(Vn+1 )|x are in fact equivalence classes of sections
% : Vn+1 → P related to each other by the condition 2
% ∼ %0 ⇔ d % − %0 |x = 0
and therefore defining one and the same horizontal distribution along the
fiber Px ;
• every section % : Vn+1 → P defined in a neighborhood of a point x and
described in coordinates as u = % (t, q 1 , . . . , q n ) may be lifted to a section
j1 (Vn+1 ) → L(Vn+1 ), denoted symbolically by %̇ , and expressed in coordi∂% k
nates as u̇ = ∂%
∂t + ∂q k q̇ . The restriction of %̇ to the fiber j1 (Vn+1 )|x is
an affine section %̇ |x : j1 (Vn+1 )|x → L(Vn+1 )|x . Two sections %, %0 satisfy
%̇ |x = %̇0 |x if and only if the differential d (% − %0 ) vanishes at x.
In view of the stated results, every element σ ∈ H(Vn+1 )|x is easily seen to determine an affine section u̇ = p 0 (σ) + pi (σ) q̇ i of the bundle L(Vn+1 )|x → j1 (Vn+1 )|x .
With the terminology of Subsection 2.2 we have thus proved
2
Notice that, consistently with the definition of H(Vn+1 ), we are not requiring %(x) = %0 (x).
12
Proposition 3.1 The affine principal fibrations (3.5), (3.6) are affine dual of
each other under the bi-affine map F : L(Vn+1 ) ×Vn+1 H(Vn+1 ) → < expressed in
coordinates as
F (t, q i , q̇ i , u̇, p 0 , pi ) = u̇ − p 0 − pi q̇ i
(3.7)
Together with Theorem 2.2, Proposition 3.1 gives rise to a canonical identification
between the first-jet spaces j1 (L(Vn+1 ), j1 (Vn+1 )) and j1 (H(Vn+1 ), Π(Vn+1 )).
In the present context, this result is further enhanced by considering the fibration
H(Vn+1 ) → < coming from the composition H(Vn+1 ) → Vn+1 → <. Denoting
by j1 (H(Vn+1 ), <) the associated first jet space, and recalling that the manifold
H(Vn+1 ) is canonically endowed with the symplectic structure (3.4), we have in
fact the following
Theorem 3.1 The manifolds j1 (H(Vn+1 ), <) , j1 (H(Vn+1 ), Π(Vn+1 )) are canonically diffeomorphic.
Proof: by definition, both manifolds in study may be regarded as affine subbundles, respectively of the tangent space T (H(Vn+1 )) and of the cotangent space
T ∗ (H(Vn+1 )) , according to the identifications
n
o
j1 (H(Vn+1 ), <)
= X | X ∈ T (H(Vn+1 )) , hX , dti = 1
(3.8a)
n
D
E
o
∂
j1 (H(Vn+1 ), Π(Vn+1 )) = ω | ω ∈ T ∗ (H(Vn+1 )) ,
(3.8b)
∂p 0 , ω = 1
The conclusion then follows from the identity
∂
∂p 0
, −X
Ω = −X ∧
∂
∂p 0
dp 0 ∧ dt + dpi ∧ dq i = X, dt
showing that the correspondence X → −X Ω determines a diffeomorphism of
j1 (H(Vn+1 ), <) onto j1 (H(Vn+1 ), Π(Vn+1 )), fibered over H(Vn+1 ) . In view of the previous results, all spaces j1 (H(Vn+1 ), <), j1 (L(Vn+1 ), j1 (Vn+1 ))
and j1 (H(Vn+1 ), Π(Vn+1 )) are canonically diffeomorphic, and may be identified.
For definiteness, and without any loss in generality, we choose to regard all of them
as different copies of the manifold j1 (H(Vn+1 ), <). Depending on the context, we
shall refer j1 (H(Vn+1 ), <) to ordinary jet coordinates t, q i , p 0 , pi , q̇ i , ṗ 0 , ṗi , or to
coordinates t, q i , u̇, pi , q̇ i , ṗ 0 , ṗi related to the previous ones by the transformation
(analogous to eq. (2.23a))
u̇ − p 0 − pi q̇ i = 0
(3.9)
The relationships with the standard jet coordinates t, q i , q̇ i , u̇, u̇ t , u̇ qi , u̇ q̇i on
j1 (L(Vn+1 ), j1 (Vn+1 )) and t, q i , pi , p 0 , p 0t , p 0qi , p 0pi on j1 (H(Vn+1 ), Π(Vn+1 )) are
then expressed by the identifications
ṗ 0 = u̇t = p 0t
,
ṗi = u̇ qi = p 0qi
,
pi = u̇ q̇i
,
q̇ i = −p 0pi
summarizing the content of eqs. (2.23b), (3.4), and of Theorem 3.1.
13
(3.10)
The quotient of j1 (H(Vn+1 ), <) by the action of the 1-parameter group of diffeomorphisms generated by the vector field ∂∂u̇ = ∂p∂ 0 will be denoted by B,
and will be referred to coordinates t, q i , q̇ i , pi , ṗ 0 , ṗi . The quotient map makes
j1 (H(Vn+1 ), <) → B into a principal fiber bundle. The Liouville 1-forms of
j1 (L(Vn+1 ), j1 (Vn+1 )) and j1 (H(Vn+1 ), Π(Vn+1 )), unified into the single expression
Θ := du̇ − ṗ 0 dt − ṗi dq i − pi dq̇ i = dp 0 − ṗ 0 dt − ṗi dq i + q̇ i dpi
(3.11)
endow j1 (H(Vn+1 ), <) → B with a canonical connection. The exterior 2-form
Υ := −d Θ = dṗ 0 ∧ dt + dṗi ∧ dq i + dpi ∧ dq̇ i
(3.12)
makes the manifold B into a symplectic manifold.
The previous discussion, summarized into the commutative diagram
j1 (H(Vn+1 ), <)
j1 (L(Vn+1 ), j1 (Vn+1 ))
?
j1 (H(Vn+1 ), Π(Vn+1 ))
?
L(Vn+1 )
(3.13)
?
H(Vn+1 )
B
PP
PP
?
j1 (Vn+1 )
PP
PP
q
)
?
Π(Vn+1 )
provides the necessary tool for the application of the Legendre transformation in
time dependent Analytical Mechanics, along the lines discussed in Section 2. An
alternative approach, leading to a construction bearing interesting analogies with
diagram (3.13) may be found in [13].
3.3
Dynamics
As a final topic, we discuss the Lagrangian and Hamiltonian formulation of Dynamics within the geometrical framework developed so far. The analysis will provide
a gauge–invariant extension to non-autonomous systems of the classical results of
Tulczyjew [5, 6, 7, 8, 9].
Let l : j1 (Vn+1 ) → L(Vn+1 ) denote a Lagrangian section, expressed in coordinates as u̇ = L(t, q i , q̇ i ). On account of the identifications (3.10), the first jet
extension j1 (l) : j1 (Vn+1 ) → j1 (L(Vn+1 ), j1 (Vn+1 )) is described by the equations
u̇ = L(t, q i , q̇ i ) ,
ṗ 0 = u̇ t =
∂L
,
∂t
ṗi = u̇ qi =
∂L
,
∂q i
pi = u̇ q̇i =
∂L
∂ q̇ i
(3.14)
The map j1 (l) carries a complete information on Dynamics. Indeed, according
to the diagram (3.13), the image space E := j1 (l)(j1 (Vn+1 )) may be viewed as
a submanifold of j1 (H(Vn+1 ), <). Switching to coordinates t, q i , p 0 , pi , q̇ i , ṗ 0 , ṗi
through eq. (3.9), let us accordingly rephrase eqs. (3.14) in the equivalent form
p0 +
∂L i
q̇ − L(t, q i , q̇ i ) = 0 ,
∂ q̇ i
ṗ 0 =
∂L
,
∂t
14
pi =
∂L
,
∂ q̇ i
ṗi =
∂L
∂q i
(3.15)
By the very definition of j1 (H(Vn+1 ), <), eqs. (3.15) provide a system of ordinary
differential equations, not in normal form, for the determination of the family of
sections γ : < → H(Vn+1 ) (⇔ γ(t) ≡ (t, q i (t), p 0 (t), pi (t)) ) whose jet extension
γ̇ := j1 (γ) satisfies γ̇(t) ∈ E ∀ t. In the resulting context, the last pair of relations
(3.15) reproduce the content of Lagrange’s equations
d ∂L
∂L
− i =0
i = 1, . . . , n
i
dt ∂ q̇
∂q
while the first pair describes the evolution of the Hamiltonian H := −L + ∂∂L
q̇ i .
q̇ i
Precisely the same state of affairs occurs if one considers a Hamiltonian section
h : Π(Vn+1 ) → H(Vn+1 ) , expressed in coordinates as p 0 = −H(t, q i , pi ). On account of eqs. (3.10), the first jet extension j1 (h) : Π(Vn+1 ) → j1 (H(Vn+1 ), Π(Vn+1 ))
is now described by the system
p 0 = −H(t, q i , pi ) ,
ṗ 0 = −
∂H
,
∂t
ṗi = −
∂H
,
∂q i
q̇ i =
∂H
∂pi
(3.16)
Once again, according to the diagram (3.13), the image space E := j1 (h)(Π(Vn+1 ))
may be regarded as a (2n + 1)–dimensional submanifold of j1 (H(Vn+1 ), <) .
Eqs. (3.16) play therefore the role of a system of ordinary differential equations,
now in normal form, characterizing the totality of sections γ : < → H(Vn+1 ) whose
jet extension satisfies γ̇(t) ∈ E ∀ t. More specifically, the last pair of eqs. (3.16)
reproduces the content of Hamilton’s equations, while the first pair describes the
evolution of the Hamiltonian.
For completeness, let us also write down the Legendre maps associated with the
sections l : j1 (Vn+1 ) → L(Vn+1 ) and h : Π(Vn+1 ) → H(Vn+1 ) considered above.
The argument is a replica of the one worked out in detail in Section 2.2, so that
we shall merely state the results:
(i) given any section l : j1 (Vn+1 ) → L(Vn+1 ) , consider the jet extension j1 (l).
Composing the latter with the (significant) vertical arrows of the diagram (3.13)
generates three maps Λl : j1 (Vn+1 ) → H(Vn+1 ) , λl : j1 (Vn+1 ) → Π(Vn+1 ) and
κl : j1 (Vn+1 ) → B.
In coordinates, expressing l as u̇ = L(t, q i , q̇ i ), we have the explicit representations
(see eqs. (2.27a, b, c))
Λl :
pi =
∂L
∂ q̇ i
λl :
pi =
∂L
∂ q̇ i
κl :
pi =
∂L
∂ q̇ i
; p 0 = L − q̇ i
∂L
∂ q̇ i
(3.17a)
(3.17b)
; ṗ 0 =
15
∂L
;
∂t
ṗi =
∂L
∂q i
(3.17c)
In view of eqs. (3.12), (3.17c), the map κl satisfies the identity
κ∗l (Υ) = κ∗l dṗ 0 ∧ dt + dṗi ∧ dq i + dpi ∧ dq̇ i =
∂L i ∂L i
∂L
dt + i dq + i dq̇ ≡ 0
=d
∂t
∂q
∂ q̇
(3.18)
indicating that the image space κl (j1 (Vn+1 )) is a Lagrangian submanifold of B.
Eqs. (3.17b) express the familiar Legendre transformation. Under the reg2
ularity assumption det ∂ q̇∂i ∂Lq̇j 6= 0, the latter may be solved with respect to
the variables q̇ i . Substituting the result into the second equation (3.17a) one
then gets the expression p 0 = −H(t, q i , pi ), describing the Hamiltonian section
h : Π(Vn+1 ) → H(Vn+1 ) associated with l.
A perfectly symmetric construction holds starting with a Hamiltonian section h : Π(Vn+1 ) → H(Vn+1 ). Once again the jet extension j1 (h) : Π(Vn+1 ) →
j1 (H(Vn+1 ), Π(Vn+1 )), composed with the significant vertical arrows of diagram
(3.13), gives rise to maps Λh : Π(Vn+1 ) → L(Vn+1 ) , λh : Π(Vn+1 ) → j1 (Vn+1 )
and κh : Π(Vn+1 ) → B, expressed in coordinates as
Λh :
q̇ i =
∂H
∂pi
λh :
q̇ i =
∂H
∂pi
κh :
q̇ i =
∂H
∂pi
; u̇ = −H +
∂H
pi
∂pi
(3.19a)
(3.19b)
; ṗ 0 = −
∂H
;
∂t
ṗi = −
∂H
∂q i
(3.19c)
H(t, q i , pi ) denoting the Hamiltonian function involved in the local representation
of h. Exactly as above, eqs. (3.12), (3.17c) provide the identity
κ∗h (Υ) = κ∗h dṗ 0 ∧ dt + dṗi ∧ dq i + dpi ∧ dq̇ i =
∂H
∂H i ∂H
=d −
dt − i dq −
dpi ≡ 0
(3.20)
∂t
∂q
∂pi
showing that the image space κh (H(Vn+1 )) is a Lagrangian submanifold of B.
2
H
Under the further assumption det ∂p∂ i ∂p
6= 0 eqs. (3.19b) may be solved with
j
respect to the variables pi , in which case the second expression (3.19a) provides
the representation of the Lagrangian section associated with h.
From a geometrical viewpoint, a significant implication of the previous discussion is the fact that, in the environment j1 (H(Vn+1 ), <) , the Lagrangian and
Hamiltonian approaches to Mechanics are nothing but different representations
of the same (2n + 1)–dimensional submanifold, described indifferently as E =
j1 (l)(j1 (Vn+1 )) = j1 (h)(Π(Vn+1 )) . This aspect is further enhanced by observing
i
that, according to eqs. (3.11), (3.14), (3.16), the embedding E −
→ j1 (H(Vn+1 ), <)
satisfies the identity
i∗ (Θ) = 0
(3.21)
16
showing that the hypersurface E is horizontal with respect to the canonical connection of j1 (H(Vn+1 ), <) → B. Now, a straightforward argument indicates that
every horizontal submanifold i : S → j1 (H(Vn+1 ), <) has dimension ≤ 2n + 1 3 .
Regular dynamical systems may therefore be viewed as horizontal submanifolds of
maximal dimension in j1 (H(Vn+1 ), <), projecting injectively onto both j1 (Vn+1 )
and Π(Vn+1 ).
The previous conclusion extends to the newer context the results originally
established by Tulczyjew in the autonomous case [5, 6, 7, 8, 9] (in this connection
see also [10, 11, 12]). The analogies are easily understood by observing that the
π
projection j1 (H(Vn+1 ), <) −
→ B sets up a 1-1 correspondence between horizontal
slicings of maximal dimension in j1 (H(Vn+1 ), <) and Lagrangian submanifolds in
B. The details are straightforward, and are left to the reader. In coordinates, the
previous assertions have their analytical counterpart in eqs. (3.18), (3.20).
π
3
Indeed, by eq. (3.21), the projection j1 (H(Vn+1 ), <) −
→ B is locally injective on S, while
eq. (3.21) itself requires i∗ (dΘ) = 0. Therefore, by the non singularity of the 2-form (3.12)
dim(S) = dim(π(S)) ≤ 12 dim(B) = 2n + 1 .
17
References
[1] E. Massa, E. Pagani and P. Lorenzoni, On the gauge structure of Classical
Mechanics, Transport Theory and Statistical Physics 29, 69–91 (2000).
[2] E. Massa, S. Vignolo and D. Bruno, Non–holonomic Lagrangian and Hamiltonian Mechanics: an intrinsic approach J. Phys. A: Math. Gen. 35, 6713–6742
(2002).
[3] E. Massa and S. Vignolo, A new geometrical framework for time–dependent
Hamiltonian Mechanics, Preprint, University of Genoa (2000).
[4] S. Vignolo, A new presymplectic geometrical framework for time–dependent
Lagrangian systems: the constraint algorithm and the second–order differential equation problem, J. Phys. A: Math. Gen. 33, 5117–5135 (2000).
[5] W. M. Tulczyjew, Hamiltonian Systems, Lagrangian Systems and the Legendre Transformation, Symposia Matematica 15, 247–258 (1974).
[6] W. M. Tulczyjew, Les sous-variété lagrangiennes et la dynamique hamiltonienne, C.R. Acad. Sci. Paris 283-A, 15–18 (1976).
[7] W. M. Tulczyjew, Les sous-variété lagrangiennes et la dynamique lagrangienne, C.R. Acad. Sci. Paris 283-A, 675–678 (1976).
[8] W. M. Tulczyjew, The Legendre Transformation, Ann. Inst. H. Poincaré 27,
101–114 (1977).
[9] W. M. Tulczyjew, Geometric Foundations of Lagrangian Mechanics, Proceedings of the I.U.T.A.M.–I.S.I.M.M. Symposium on Modern Developments in
Analytical Mechanics, Academy of Sciences of Turin, Turin, June 7–11, (1982),
Atti dell’Accademia delle Scienze di Torino, Suppl. Vol. 117, 255–272 (1983).
[10] S. Benenti, Symplectic Relations in Analytical Mechanics, Proceedings of the
I.U.T.A.M.–I.S.I.M.M. Symposium on Modern Developments in Analytical
Mechanics, Academy of Sciences of Turin, Turin, June 7–11, (1982), Atti
dell’Accademia delle Scienze di Torino, Suppl. Vol. 117, 39–91 (1983).
[11] S. Benenti, Relazioni simplettiche, Pitagora Editrice, Bologna, (1988).
[12] J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories,
Lecture Notes in Physics, Vol. 107, Springer, Berlin (1979).
[13] W. M. Tulczyjew and P. Urbanski, An affine framework for the dynamics
of charged particles, in: “La Mécanique Analytique de Lagrange et son Heritage – II”, Atti dell’Accademia delle Scienze di Torino, Suppl. n. 2, Vol. 126,
257–65 (1992).
18