arXiv:1405.0748v2 [math-ph] 27 May 2014arXiv:1405.0748v2 [math-ph] 27 May 2014 TULCZYJEW’S APPROACH FOR PARTICLES IN GAUGE FIELDS GUOWU MENG ABSTRACT.In mid-1970s W. M. Tulczyjew

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4 TULCZYJEW’S APPROACH FOR PARTICLES IN GAUGE FIELDS

GUOWU MENG

ABSTRACT. In mid-1970s W. M. Tulczyjew discovered an approach to classical mechan-ics which brings the Hamiltonian formalism and the Lagrangian formalism under a com-mon geometric roof: the dynamics of a particle with configuration spaceX is determinedby a Lagrangian submanifoldD of TT ∗X (the total tangent space ofT ∗X), and the de-scription ofD by its HamiltonianH: T ∗X → R (resp. its LagrangianL: TX → R)yields the Hamilton (resp. Euler-Lagrange) equation.

It is reported here that Tulczyjew’s approach also works forthe dynamics of (charged)particles in gauge fields, in which the role of the total cotangent spaceT ∗X is playedby Sternberg phase spaces. In particular, it is shown that, for a particle in a gauge field,the equation of motion can be locally presented as the Euler-Lagrange equation for a La-grangian which is the sum of the ordinary LagrangianL(q, q̇), the Lorentz term, and anextra new termwhich vanishes whenever the gauge group is abelian. A chargequanti-zation condition is also derived, generalizing Dirac’s charge quantization condition fromU(1) gauge group to any compact connected gauge group.

CONTENTS

1. Introduction 22. Sternberg phase space 32.1. Principal connection 42.2. Sternberg formΩΘ onF 42.3. Sternberg phase space 63. A canonical isomorphism of double vector bundles 63.1. Local formulae 94. Special symplectic manifold 105. Tulczyjew triple and its magnetized version 115.1. Classical Tulczyjew triple 115.2. Magnetized Tulczyjew triple 126. Tulczyjew’s approach for particles in gauge fields 156.1. The Hamiltonian formulation 156.2. The Lagrangian formulation 166.3. The Legendre transformation 19Appendix A. Useful facts 20A.1. Tangent lift [18] 20A.2. “Cotangent map”T ∗f 20Appendix B. A list of symbols 22References 23

Date: May 1, 2014.The author is supported by Hong Kong University of Science and Technology under DAG S09/10.SC02.

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http://arxiv.org/abs/1405.0748v2

2 GUOWU MENG

1. INTRODUCTION

Tulczyjew’s unified approach [1] to Lagrangian and Hamiltonian descriptions of particledynamics, though not well-known outside a small circle of mathematicians and physicists,is quite appealing to geometry-oriented minds. In this approach, the Legendre transfor-mation takes a specially simple interpretation, and systems with singular Lagrangians orsubject to constraints appear naturally. An advantage of this geometric approach is its flex-ibility in the sense that it can be easily adapted to different settings. As one more demon-stration of its flexibility, in this article it is shown that Tulczyjew’s approach also works forparticle dynamics in which a charged particle moves in the presence of an external gaugefield, either abelian or non-abelian.

The incorporation of a gauge field into the classical particle dynamics is a nontrivialbusiness when the gauge group is non-abelian. It seems that,in the setting of symplecticgeometry, this was initially done by S. Sternberg [2]. Some further elaborations, especiallythe one on the relationship of Sternberg’s symplectic approach with the earlier Poissonapproach of S. K. Wong [3], came from A. Weinstein [4] and R. Montgomery [5].

For readers who are familiar with the notion of double vectorbundle in the sense ofPradines [6] (or equivalently in the sense of Grabowski and Rotkiewicz [7]) and closely-related notion of double Lie algebroid in the sense of K. Mackenzie [8] (or equivalentlyin the sense of T. Voronov [9]), it is worth to remark that, just as Tulczyjew’s originalapproach to particle dynamics, our extension to dynamics ofcharged particle rests on thefollowing mathematical fact [10]:a real vector bundleE → X and its dual vector bun-dleE∗ → X are not canonically isomorphic, but their associated double vector bundles(T ∗E∗;E∗, E∗∗;X) and(T ∗E;E∗, E;X) are.

All ingredients involved in the present work have already appeared in the literature, butthey will be reviewed here for completeness. It should be pointed out that, the main resultobtained here would appear earlier (especially in Ref. [11]) if enough attention was paidto Sternberg phase spaces. After communicating with J. Grabowski and P. Urbański, theauthor learned that there had been an approach to electrically charged particles based onaffine geometries [12, 13, 14]; this is associated withR-principal bundles, but a reasonableextension to non-abelian gauge fields may also be possible.

In Section 2 a detailed review of Sternberg phase space [2] ispresented. For ourpurpose, the presentation given here focuses more on the explicit local computations.In Section 3 a detailed review of the canonical isomorphismT ∗E∗ ∼= T ∗E for anyreal vector bundleE → X is presented. This isomorphism is more natural when itis viewed as the canonical double vector bundle isomorphismfrom (T ∗E∗;E∗, E∗∗;X)onto (T ∗E;E∗, E;X); moreover, it enables Tulczyjew to bring the Lagrangian descrip-tion of classical mechanics to the domain of symplectic geometry and unify Lagrangianand Hamiltonian formalisms of classical mechanics under a single geometric roof. In Sec-tion 4 we review Tulczyjew’s notion ofspecial symplectic manifold, a concept more refinedthan that of symplectic manifold. In Section 5 we first reviewthe classical Tulczyjew tripleused in the Tulczyjew’s approach to particle dynamics and then introduce its magnetizedversion where Sternberg phase spaces play the role ofT ∗X . Section 6 contains new re-sults. Here, the Hamiltonian side of Tulczyjew’s approach,taken from Ref. [1], works forarbitrary symplectic manifoldM , not just for the Sternberg phase space. The magnetizedversion of Tulczyjew triple introduced in Section 5 enablesus to work out the Lagrangianside of Tulczyjew’s approach to particles in gauge fields. Inparticular, for particles inYang-Mills fields, a Lagrangian approach in the usual textbook way is derived, for which

TULCZYJEW’S APPROACH FOR PARTICLES IN GAUGE FIELDS 3

the Lagrangian is the sum of the ordinary LagrangianL(q, q̇), the Lorentz term, and anextra new term. A generalized version of Dirac’scharge quantization condition[15] isalso derived from this Lagrangian approach. (Please consult the Main Theorem on page17 for precise statement.) As far as this author knows, the aforementioned Lagrangian andcharge quantization condition are new; cf. Ref. [19] and example 2 on page 18.

For the convenience of readers, in the appendix we list some symbols used in this article,along with technically useful facts on tangent lift operator and on fiber bundle equippedwith a connection. Most of symbols used in this article are quite standard in the math-ematical literature, for example,τX : TX → X denotes the tangent bundle ofX , πX :T ∗X → X denotes the cotangent bundle ofX , ϑX denotes the Liouville form onT ∗XandωX := dϑX denotes the tautological symplectic form onT ∗X . Also, for notationalsanity in this article, we shall use the same notation for both a differential form (or a map)and its pullback under a fiber bundle projection map. For example, for a differential formΩ on manifoldF , its pullback under projectionX × F → F is also denoted byΩ.

Acknowledgements. The author learned Tulczyjew’s elegant approach from JanuszGrabowski at the recent workshop on Geometry of Mechanics and Control Theory (IndiaInstitute of Sciences, Bangalore, India, January 2 - 10, 2014). Besides thanking JanuszGrabowski for his beautiful talk, he would also like to thankPartha Guha for organizingthe wonderful workshop. Finally, he would like to thank JohnBaez, Janusz Grabowski,Jim Stasheff and Pawel Urbański for providing either valuable comments or additionalreferences.

2. STERNBERG PHASE SPACE

Throughout this section we assume thatX is a manifold,G is a compact connected Liegroup with Lie algebrag,P → X is a principalG-bundle with a fixed principal connectionformΘ, andF is a hamiltonianG-space with symplectic formΩ and (equivariant) momentmapΦ. Recall that we shall use the same notation for both a differential form (or a map)and its pullback under a fiber bundle projection map. For example, the pullback of themoment mapΦ under projectionX × F → F is also denoted byΦ.

Let F := P ×G F andF ♯ be the pullback of diagram

F

T ∗X XπX

Sternberg observed that [2], with the above data, there is a correct substituteΩΘ onF forΩ onX×F , in the sense thatΩΘ is a closed real differential two-form onF and it is equalto Ω whenP → X is a trivial bundle with the product connection. He further observedthat, ifωX denote the canonical symplectic form onT ∗X , then

ωX +ΩΘ

is a symplectic form onF ♯ — theSternberg symplectic form.

To describe the Sternberg symplectic form, we need to do somepreparations. Fora ∈G, the right action ofa onP is denoted byRa and the adjoint action ofa ong is denotedbyAda. Forξ ∈ g, the infinitesimal right action ofξ onP is a vector field onP and shallbe denoted byXξ. SinceG is a compact connected Lie group, we can assume that it is a

4 GUOWU MENG

Lie subgroup ofSO(N) for some positive integerN . Let us denote byg the inclusion mapof G into the vector space of all real square matrices of orderN . Note that, in terms ofg,the Maurer-Cartan form onG can be written asg−1 dg, here, the product betweeng−1 anddg is the matrix multiplication.

2.1. Principal connection. Let Θ be ag-valued differential one-form onP . ThenΘis/defines a principal connection on the principalG-bundleP → X if it satisfies the fol-lowing two conditions1:

1)Ra−1∗ Θ = AdaΘ for anya ∈ G, 2)Θ(Xξ) = ξ for anyξ ∈ g.

Working locally, we may assume thatP → X is trivial. Suppose that

X ×G P

M

φ

∼=

is a trivialization, then, as ag-valued differential one-form onX ×G,

φ∗Θ = g−1Aφg + g−1 dg(2.1)

for a uniqueg-valued differential one-formAφ onX . Similarly, if φ′ is another trivializa-tion, we have

φ′∗Θ = g−1Aφ′g + g

−1 dg

for a uniqueg-valued differential one-formAφ′ onX . To see howAφ andAφ′ are related,we note that the bundle isomorphismλ defined by the commutative triangle

P

X ×G X ×G

φ

λ

φ′

can be written asλ(x, b) = (x, a(x)b) for a unique smooth mapa: X → G. Sinceφ∗Θ = λ∗φ′

∗Θ, we have

Aφ = a−1Aφ′a+ a

−1 da or Aφ′ = aAφa−1 + a da−1 .(2.2)

2.2. Sternberg form ΩΘ on F . For the hamiltonianG-spaceF , recall thatΩ is its sym-plectic form andΦ: F → g∗ is its moment map. We letYξ be the vector field onFwhich represents the infinitesimal left action ofξ ∈ g onF . ThenΩ is invariant under theG-action onF , Φ isG-equivariant, and

YξyΩ = 〈ξ, dΦ〉 for anyξ ∈ g.(2.3)

Here y denotes the interior product and〈 , 〉 denotes the paring of elements ing withelements ing∗. In view of the fact thatΦ isG-equivariant, Eq. (2.3) implies that

Ω(Yξ1 , Yξ2) = 〈[ξ1, ξ2],Φ〉 for anyξ1, ξ2 ∈ g.(2.4)

1OurΘ here is the negative of theΘ in Ref. [2] becauseRa is right multiplication bya here and bya−1 inRef. [2].


Denote byφF the composition mapX ×F ∼= (X ×G)×G Fφ×GF−→ P ×G F =: F . If

we letλF be the fiber bundle isomorphism in the commutative triangle

F

X × F X × F,λF

φF φ′

F

thenλF (x, f) = (x, a(x) · f). Let

Ωφ := Ω− d〈Aφ,Φ〉 .

The following lemma implies that there is a well-defined closed real differential two-formΩΘ onF , referred to as theSternberg form onF , such thatΩφ = φ∗FΩΘ.

Lemma 2.1. With the notations as above, we haveλ∗F Ωφ′ = Ωφ. Consequently, as closedreal differential two-forms onF , (φF

−1)∗Ωφ = (φ′F−1

)∗Ωφ′ .

SinceΩφ′ = Ω− d〈Aφ′ ,Φ〉 andΩφ = Ω− d〈Aφ,Φ〉, we have

λ∗FΩφ′ = λ∗FΩ− d〈Aφ′ , L

∗aΦ〉 hereLa is the left action ofa onF

= λ∗FΩ− d〈Aφ′ ,Ada−1∗Φ〉 becauseΦ isG-equivariant

= λ∗FΩ− d〈Ada−1Aφ′ ,Φ〉= λ∗FΩ− d〈Aφ,Φ〉+ d〈a

−1da,Φ〉 using Eq. (2.2)= Ωφ + λ

∗FΩ− Ω+ d〈a

−1da,Φ〉,

so the above lemma is equivalent to

Claim. LetλF : X × F → X × F , Φ: F → g∗, a: X → G, andΩ be as before. Thenλ∗FΩ = Ω− d〈a

−1da,Φ〉.

Proof. Let ξ = da a−1. SinceλF (x, f) = (x, a(x) · f), we have

T(x,f)λF : TxX × TfF → TxX × Ta(x)·fF(u, v) 7→ (u, TfLa(x)(v) + (Yuyξ|x)|a(x)·f ).

With the understanding thatΩ represents both the symplectic form onF and its pullbackunder projectionX × F → F , for (u1, v1), (u2, v2) in T(x,f)(X × F ),

(λ∗FΩ)|(x,f)((u1, v1), (u2, v2))

is equal to

(L∗a(x)Ω)|f (v1, v2)

+Ω(Yu1y ξ|x , Yu2y ξ|x)|a(x)·f+Ω|a(x)·f(Yu1y ξ|x |a(x)·f , TfLa(x)(v2))− Ω|a(x)·f (Yu2y ξ|x |a(x)·f , TfLa(x)(v1)).

Let η = a−1 da. Then, in view of the fact thatΦ is G-equivariant andη = ada−1ξ, theabove expression becomes

Ω|f (v1, v2) becauseΩ isG-invariant+〈[u1y η|x, u2y η|x],Φ|f 〉 using Eq. (2.4)+〈u1y η|x, v2y dΦ|f〉 − 〈u2y η|x, v1y dΦ|f〉 using Eq. (2.3)

= Ω|f (v1, v2) + 〈η2|x,Φ|f 〉(u1, u2) + 〈η|x, dΦ|f 〉((u1, v1), (u2, v2))

= (Ω− d〈η,Φ〉)|(x,f)((u1, v1), (u2, v2)) becausedη = −η2.

Thereforeλ∗FΩ = Ω− d〈η,Φ〉 = Ω− d〈a−1da,Φ〉. �

6 GUOWU MENG

2.3. Sternberg phase space.With the Sternberg formΩΘ onF and the canonical sym-plectic formωX on T ∗X being both closed, and the fact from the definition ofF ♯ thatthere are fiber bundle projections fromF ♯ toF and also toT ∗X , we know that

ωΘ := ωX +ΩΘ(2.5)

is a closed real differential two-form onF ♯.

Claim. With the notations as above,ωΘ is non-degenerate everywhere onF ♯, so it is asymplectic form onF ♯.

Proof. Introducing local coordinate functions(qi, pj) onT ∗X andzα onF , and denoting∂∂qi

by ∂i, ∂∂zα by ∂α, thenωΘ can be locally represented by

dpi ∧ dqi +

1

2Ωαβ dz

α ∧ dzβ −1

2〈∂iAj − ∂jAi,Φ〉dq

i ∧ dqj + 〈Ai, ∂αΦ〉dqi ∧ dzα

which is then easy to see to be non-degenerate everywhere. Therefore,ωΘ is a symplecticform onP ♯. �

In summary, we have

Theorem 2.2(Sternberg, 1977). With the data and notations given in the beginning of thissection, we have the following statements.

1) There is a closed real differential two-formΩΘ onF which is of the formΩ−d〈A,Φ〉under a local trivialization ofP → X in which the connection formΘ is represented bytheg-valued differential one-formA onX .

2) The differential two-formωΘ := ωX +ΩΘ is a symplectic form onF ♯.

The symplectic manifold(F ♯, ωΘ) is referred to as aSternberg phase space. In par-ticular (T ∗X,ωX) is a Sternberg phase space. In Ref. [4] A. Weinstein introduced asymplectic space out of the principalG-bundleP → X and the hamiltonianG-spaceF ,and showed that a connectionΘ onP → X yields a symplectomorphism from his sym-plectic space to the Sternberg phase space(F ♯, ωΘ), a reason for A. Weinstein to call hissymplectic space theuniversal phase space.

3. A CANONICAL ISOMORPHISM OF DOUBLE VECTOR BUNDLES

The purpose of this section is to give a detailed review of thefollowing simple andelegant mathematical fact [10]:a real vector bundleE → X and its dual vector bundleE∗ → X are not canonically isomorphic, butT ∗E∗ andT ∗E are canonically isomorphicas symplectic manifolds.

Let π: E → X be a real vector bundle andπ∗: E∗ → X be its dual vector bundle.Consider the diagram

T ∗X E∗

E X

πXπ∗

π

(3.1)

The limit of diagram (3.1) exists and is unique up to diffeomorphisms, in fact, it is thetotal space of the Whitney sum of the three vector bundles over X . We shall show thatbothT ∗E∗ andT ∗E can be this limit, so they must be diffeomorphic to each other. Thedetailed arguments are given below.


Step one. For anye ∈ E, we have injective linear mapEπ(e) ∼= TeEπ(e) ⊂ TeE whosedual is a surjective linear mapT ∗eE → E

∗π(e) which, upon being globalized, becomes the

top arrowpE in the commutative square

T ∗E E∗

E X.

pE

πE π∗

π

(3.2)

Step two. Choosing a connection onE → X , then we have the commutative square

T ∗E T ∗X

E X

T∗π

πE πX

π

by appendix A.Step three. Combining steps one and two, we have commutative diagram

T ∗E

T ∗X E∗

E X

pE

πE

T∗π

πXπ∗

π

(3.3)

which in turn yields a smooth mapT ∗E → T ∗X ⊕ E ⊕ E∗, fibering overX . Thissmooth map is a bijection becauseT ∗e E → T

∗π(e)X × {e} × E

∗π(e) is a bijection for each

e ∈ E. Moreover, by using the local triviality ofEπ→ X , one can check that this map is

a diffeomorphism, so it turnsT ∗E → X into a vector bundle overX . This vector bundleis isomorphic toT ∗X ⊕ E ⊕ E∗, but it is not canonical because of its dependence on thechoice of a connection onE

π→ X .

Step four. SinceE∗∗ ∼= E naturally and a connection onEπ→ X is turned into a

connection onE∗π∗

→ X upon taking dual, replacingE by E∗ in the above analysis, com-mutative diagram (3.3) becomes commutative diagram

T ∗E∗

T ∗X E∗∗ ∼= E

E∗ X

pE∗

πE∗

T∗π∗

πXπ∗∗

π∗

So we have vector bundle isomorphismT ∗E∗ ∼= T ∗X ⊕E∗ ⊕E∗∗ by the same argumentas above.

8 GUOWU MENG

Step five. Defineκ via commutative diagram

T ∗E∗ T ∗X ⊕ E∗ ⊕ E∗∗

T ∗E T ∗X ⊕ E∗ ⊕ E

T∗π∗ ⊕ πE∗ ⊕ pE∗

∼=

κ ∼= 1 ⊕ 1 ⊕ −ι−1∼=

T∗π ⊕ pE ⊕ πE∼=

(3.4)

where−ι−1 is the negative of the inverse of the natural vector bundle identification

ι : E −→ E∗∗

u 7→ α 7→ 〈u, α〉.(3.5)

We shall see later that,κ is a symplectomorphism and is independent of the choice of aconnection onE → X .

In summary, once a connection onEπ→ X is chosen,T ∗E andT ∗E∗ become vector

bundles overX ; moreover, we have vector bundle isomorphismsT ∗E ∼= T ∗X ⊕ E ⊕ E∗

andT ∗E∗ ∼= T ∗X ⊕ E∗ ⊕ E∗∗. SinceE ⊕ E∗ ∼= E∗ ⊕ E∗∗, we have vector bundleisomorphismκ: T ∗E∗ → T ∗E overX . While the vector bundle structures onT ∗E → XandT ∗E∗ → X depend on the choice of a connection onE

π→ X , the diffeomorphismκ

does not if we identifyE ⊕ E∗ with E∗ ⊕ E∗∗ via(

0 1−ι 0

)

whereι is the map defined in Eq. (3.5). SinceπE∗ : T ∗E∗ → E∗ is a vector bundle andthe definition ofκ in diagram (3.4) makes triangle

T ∗E∗ T ∗E

E∗

κ∼=

πE∗ pE

commutative, the canonical diffeomorphismκ turnspE : T ∗E → E∗ into a canonicalvector bundle so thatκ becomes a canonical vector bundle isomorphism overE∗ and(T ∗E;E,E∗;X) as in diagram (3.2) becomes a (canonical)double vector bundle(withT ∗X

πX→ X as its core) in the sense of J. Pradines [6].In short,to any real vector bundleE

π→ X , there associate two canonically isomorphic

double vector bundles(T ∗E;E,E∗;X) and(T ∗E∗;E∗, E∗∗;X):

T ∗E∗ E∗∗

E∗ X

T ∗E E

E∗ X

πE∗

pE∗

π∗∗

−ι−1π∗

1

pE

πE

ππ∗

1

where the dashed arrow isκ. Moreoverκ is a symplectomorphism.This isomorphism ofdouble vector bundles shall be referred to asthe canonical isomorphism.


It remains to show thatκ is a canonical symlectomorphism, i.e., it is a symlectomor-phism and is independent of the choice of a connection onE

π→ X . That will be clear after

we work out a local representation forκ.

3.1. Local formulae. Let n = dimX andk be the rank ofE π→ X . Since we worklocally, we may assume thatX is diffeomorphic toRn andE

π→ X is trivial. Let us fix a

diffeomorphismQ: X → Rn and a trivialization

E X × Rk

X

φ

∼=

π p1

wherep1 is the projection onto the first factor. Then we have diffeomorphisms

E ∼= X × Rk ∼= Rn × Rk, E∗ ∼= X × (Rk)∗ ∼= Rn × Rk

which shall be denoted by(q, u) and(q, α) respectively, and diffeomorphisms

T ∗E ∼= Rn × Rk × (Rn)∗ × (Rk)∗ ∼= (Rn × Rk)2,T ∗E∗ ∼= Rn × (Rk)∗ × (Rn)∗ × (Rk)∗∗ ∼= (Rn × Rk)2

which shall be denoted by(q, u, p, α) and(q, α, p, û) respectively.Under the trivializationφ, the connection onE

π→ X is represented by a realk × k-

matrix valued differential one-formA · dq := Ai dqi on X , according to formula (A.3),

the mapT ∗ET∗π→ T ∗X can be represented by

(q, u, p, α) 7→ (q, p− α ·Au)(3.6)

where· means the dot product ofRk, andA is viewed as a local function onT ∗Y . On

the other hand, the connection onE∗π∗

→ X is represented by−AT onX , so, according to

formula (A.3), the mapT ∗E∗T∗π∗→ T ∗X can be represented by

(q, α, p, û) 7→ (q, p+ û ·ATα) = (q, p+ α ·Aû)).(3.7)

Note also that, the mapT ∗EπE→ E andT ∗E

pE→ E∗ are represented by

(q, u, p, α) 7→ (q, u), (q, u, p, α) 7→ (q, α)

respectively, the mapT ∗E∗πE∗→ E∗ andT ∗E∗

pE∗→ E∗∗ are represented by

(q, α, p, û) 7→ (q, α), (q, α, p, û) 7→ (q, û)

respectively, andι: E → E∗∗ is represented by(q, u) 7→ (q, u). Therefore, from thedefinition ofκ in diagram (3.4), we conclude that Tulczyjew isomorphismκ: T ∗E∗ →T ∗E is represented by

(q, α, p, û) 7→ (q,−û, p, α).(3.8)

This local representation ofκ immediately implies thatκ preserves the natural symplec-tic structures, is independent of the choice of connection on E

π→ X , and fibers over both

E∗ andE, i.e., both triangle

T ∗E∗ T ∗E

E∗

κ∼=

πE∗ pE

10 GUOWU MENG

and square

T ∗E∗ T ∗E

E∗∗ E

κ∼=

pE∗ πE

−ι−1

∼=

are commutative, also an easy fact from commutative diagram(3.4).

4. SPECIAL SYMPLECTIC MANIFOLD

In mid-1970s W. M. Tulczyjew discovered an approach to classical mechanics whichbrings the Hamiltonian formalism and the Lagrangian formalism under a common geo-metric roof: the dynamics of a particle with configuration spaceX is determined by aLagrangian submanifoldD of TT ∗X (the total tangent space ofT ∗X), and the descrip-tion ofD by its HamiltonianH : T ∗X → R (resp. its LagrangianL: TX → R) yields theHamilton (resp. Euler-Lagrange) equation.

To formulate this approach to mechanics, Tulczyjew introduced the notion ofspecialsymplectic manifold. He observed that, onTT ∗X , there is one symplectic manifold struc-ture and two special symplectic manifold structures (refereed to asLiouville structuresinRef. [16, 17]); therefore, for a classical particle with configuration spaceX under a givenconservative force, there is one dynamics (i.e., the submanifold D which is Lagrangianwith respect to the symplecic structure onTT ∗X) and two descriptions of this dynamics(i.e., the description ofD via the two Liouville structures onTT ∗X).

Let (P, ω) be a symplectic manifold with symplectic formω, N be a submanifold ofP .We say thatN is anisotropic submanifoldof (P, ω) if the pullback ofω under the inclusionN →֒ P is identically zero, and is anLagrangian submanifoldof (P, ω) if it is isotropicanddimP = 2dimN .

Definition 4.1. A special symplectic manifoldis a quadruple(P,M, π, ϑ), where(P,M, π)is a smooth fiber bundle,ϑ is a differential one-form onP , and there is a diffeomorphismα: P → T ∗M such thatπ = πM ◦ α andϑ = α∗ϑM .

The diffeomorphismα is unique, assuming it exists. If(P,M, π, ϑ) is a special sym-plectic manifold, then(P, dϑ) is a symplectic manifold isomorphic to(T ∗M,ωM ), and iscalled theunderlying symplectic manifold of (P,M, π, ϑ).

The following proposition is obvious.

Proposition 4.1. 1) Let (M,ω) be a symplectic manifold. Then(TM,M, τM , iTω) is aspecial symplectic manifold.

2) Let(P1,M1, π1, ϑ1) and(P2,M2, π2, ϑ2) be special symplectic manifolds. Then thequadruple(P2 × P1,M2 ×M1, π2 × π1, ϑ2 − ϑ1) is a special symplectic manifold.

The special symplectic manifold in part 1) is referred to as aHamiltonian specialsymplectic manifold. For us, the interestingM is T ∗X or more generally a Sternbergphase space.

The following two propositions are taken from Ref. [20, Section 3].

Proposition 4.2. Let (P,M, π, ϑ) be a special symplectic manifold,K a submanifold ofM andF a smooth real function onK. Then the set

N := {p ∈ π−1(K) | ι∗π−1(K)ϑ = (π|π−1(K)∗dF at p}


is a Lagrangian submanifold of(P, dϑ), K = π(N), the mapping̺ defined by the com-mutative diagram

N P

K M

ιN

̺ π

ιK

is a submersion, the fibers of̺ are connected andι∗Nϑ = ̺∗dF .

The Lagrangian submanifoldN in this proposition is called the Lagrangian submanifoldgeneratedbyF , andF is called agenerating functionof N .

Proposition 4.3. Let (P,M, π, ϑ) be a special symplectic manifold,K a submanifold ofM andN an isotropic submanifold of(P, dϑ) such thatK := π(N) is a submanifold ofM , the mapping̺ defined by the commutative diagram

N P

K M

ιN

̺ π

ιK

is a submersion and the fibers of̺ are connected. Then there is a unique closed differentialformγ onK such thatι∗Nϑ = ̺

∗γ. If γ is exact andγ = dF , thenN is contained in theLagrangian submanifold generated byF , and ifN is a Lagrangian submanifold generatedby a functionF , thendF = γ.

5. TULCZYJEW TRIPLE AND ITS MAGNETIZED VERSION

We shall split the discussion of the Tulczyjew triple into two parts: the classical Tulczy-jew triple and the magnetized Tulczyjew triple, though the former is a special case of thelater.

5.1. Classical Tulczyjew triple. Since(T ∗X,ωX) is a symplectic manifold, there is avector bundle isomorphismβX : TT ∗X → T ∗T ∗X overT ∗X . LetαX = κ ◦ βX , whereκ: T ∗T ∗X → T ∗TX is the canonical isomorphism reviewed in Section 3, then we haveTulczyjew triple

T ∗T ∗X TT ∗X T ∗TX.βX∼=

αX∼=

(5.1)

Let

ϑHX := β∗XϑT∗X , ϑ

LX := α

∗XϑTX .(5.2)

SinceT ∗X is a symplectic manifold, by Proposition 4.1, we have a Hamiltonian specialsymplectic manifold

SHX := (TT∗X,T ∗X, τT∗X , ϑ

HX)

as usual. With the help of local formula (5.4), one can check that diagram

TT ∗X T ∗TX

TX TX

αX

TπX πTX

1

(5.3)

12 GUOWU MENG

is commutative, so quadruple

SLX := (TT∗X,TX, TπX, ϑ

LX)

is a special symplectic manifold. We shall callSLX a Lagrangian special symplecticmanifold.

5.1.1. Local formulae.In Section 3, if we takeE → X to beTX → X , then the canonicalisomorphismκ: T ∗T ∗X → T ∗TX has this local representation:

(q, p, ṗ, q̇) 7→ (q,−q̇, ṗ, p),

so we have

βX(q, p, q̇, ṗ) = (q, p, ṗ,−q̇), αX(q, p, q̇, ṗ) = (q, q̇, ṗ, p)(5.4)

in local representation. Therefore, locallyϑHX = ṗi dqi − q̇i dpi, ϑLX = ṗi dq

i + pi dq̇i. In

more compact form, we have

ϑHX = ṗ · dq − q̇ · dp, ϑLX = ṗ · dq + p · dq̇(5.5)

locally. In view of the fact that locallŷϑX = p · q̇, we have

ϑLX − ϑHX = dϑ̂X .(5.6)

Then dϑLX = dϑHX =: ΩX , so ϑ

LX and ϑ

HX yields the same symplectic formΩX on

TT ∗X . Note that locally

ΩX = dṗi ∧ dqi + dpi ∧ dq̇

i.(5.7)

In terms of operatorsiT anddT on page 20 for tangent bundleTT ∗X → T ∗X , we have

ϑ̂X = iTϑX , ϑHX = iTωX , ϑ

LX = dTϑX , ΩX = dTωX(5.8)

whose validity can be checked by local computations based onformulae on page 20.

In summary, there are two special symplectic structures onTT ∗X that underlie thesymplectic structureΩX on TT ∗X , the one that corresponds toSHX is called theHamil-tonian special symplectic structure, and the one that corresponds toSLX is called theLagrangian special symplectic structure. When we go fromT ∗X to a generic symplec-tic manifoldM , the Hamiltonian special symplectic structure still exists (onTM ) as usual,but the Lagrangian special symplectic structure ceases to exist. A key observation of thisarticle is thatthe Lagrangian special symplectic structure still exist ifwe go fromT ∗X toa Sternberg phase space.

5.2. Magnetized Tulczyjew triple. In this subsection we shall assume thatG is a com-pact connected Lie group,P

p→ X is a principalG-bundle with a fixed principal con-

nection formΘ, Fρ→ X is the associated fiber bundle with fiberF and the associated

G-connection. We further assume thatF is a hamiltonianG-space with (G-equivariant)moment mapΦ.


Denote byF♯ andF ♯ the manifolds defined in the pullback diagrams

F♯ F

TX X,

τ̃X

ρ♯ ρ

τX

F ♯ F

T ∗X X

π̃X

ρ♯ ρ

πX

(5.9)

respectively. We note that the dual vector bundle ofF♯τ̃X→ F is F ♯

π̃X→ F , so we have thecanonical isomorphism

κ : T ∗F ♯ → T ∗F♯

as demonstrated in Section 3. In view of the fact that Sternberg phase spaceF ♯ is a sym-plectic manifold, we have a generalized Tulczyjew triple:

TF ♯

T ∗F ♯ T ∗F♯

αFβF

κ

(5.10)

where isomorphismβF comes from the symplectic structure onF ♯ andαF := κ ◦ βF .This generalized Tulczyjew triple shall be referred to as magnetized Tulczyjew triple.

Let

ϑHF

:= β∗FϑF♯ , ϑ

LF

:= α∗FϑF♯(5.11)

andϑ̂X also denote the pullback of̂ϑX under mapTρ♯: TF ♯ → TT ∗X . Later we shallshow that

ϑLF

− ϑHF

= dϑ̂X .(5.12)

ThendϑLF

= dϑHF

=: ΩF , soϑLF andϑHF

yield the same symplectic formΩF onTF ♯.We shall callϑL

F(resp. ϑH

F) the Lagrangian (resp. Hamiltonian ) Liouville form on

TF ♯.

5.2.1. Local formulae.To get a local formula forβF , we need to get a local formulafor the symplectic form onF ♯. Since we work locally we may assume thatP → X isX × G → X and then the connection formΘ is equal tog−1Ag + g−1 dg whereA isa g-valued differential one-form onX andg−1 dg is the Maurer-Cartan form onG. TheSternberg symplectic formωΘ onT ∗X × F is

ωX + Ω− d〈A,Φ〉

which, in local coordinates, is

dpi ∧ dqi +

1

2Ωαβ dz

α ∧ dzβ −1

2〈∂iAj − ∂jAi,Φ〉dq

i ∧ dqj + 〈Ai, ∂αΦ〉dqi ∧ dzα.

ThenβF : TF ♯ → T ∗F ♯ can be represented as

(q, p, z, q̇, ṗ, ż) 7→ (q, p, z, ṗi − 〈q̇j(∂jAi − ∂iAj),Φ〉 − 〈Ai, ż

α∂αΦ〉,

−q̇, żαΩαβ + 〈q̇iAi, ∂βΦ〉).(5.13)

ConsequentlyαF : TF ♯ → T ∗F♯ can be represented as

(q, p, z, q̇, ṗ, ż) 7→ (q, q̇, z, ṗi − 〈q̇j(∂jAi − ∂iAj),Φ〉 − 〈Ai, ż

α∂αΦ〉,

pj , żαΩαβ + 〈q̇

iAi, ∂βΦ〉).(5.14)

14 GUOWU MENG

Then locallyϑHF

is equal to

(ṗi − 〈q̇j(∂jAi − ∂iAj),Φ〉 − 〈Ai, ż

α∂αΦ〉) dqi − q̇i dpi + (ż

αΩαβ + 〈q̇iAi, ∂βΦ〉) dz

β.

Similarly, locallyϑLF

is equal to

(ṗi − 〈q̇j(∂jAi − ∂iAj),Φ〉 − 〈Ai, ż

α∂αΦ〉) dqi + pi dq̇

i + (żαΩαβ + 〈q̇iAi, ∂βΦ〉) dz

β.

Since locallyϑ̂X = piq̇i, identity (5.12) is verified, as promised.In terms of operatorsiT anddT on page 20 for tangent bundleTF ♯ → F ♯, we have

ϑHF

= iTωΘ,

ϑLF

= dTϑX + iTΩΘ,

ΩF = dTωΘ

(5.15)

whose validity can be checked by local computations based onformulae on page 20. NotethatϑX is really the pullback ofϑX underF ♯ → T ∗X andΩΘ is really the pullback ofΩΘ underF ♯ → F .

5.2.2. Special symplectic structures onTF ♯. SinceF ♯ is a symplectic manifold, by Propo-sition 4.1, we have aHamiltonian special symplectic manifold

SHF

:= (TF ♯,F ♯, τF♯ , ϑHF)

as usual.Since diagram (5.3) is commutative, diagram

TF ♯ F ♯ F

TT ∗X T ∗X X

TX

τF♯

Tρ♯

τ̃X

ρ♯ ρ

τT∗X

TπX

πX

τX

is commutative, so we have a smooth mapTF : TF ♯ → F♯. LocallyTF can be representedas follows:

(q, p, z, q̇, ṗ, ż) 7→ (q, q̇, z).(5.16)

This local formulae, together with local formula (5.14), implies that that diagram

TF ♯ T ∗F♯

F♯ F♯

αF

TF πF♯

1

is commutative. So quadruple

SLF

:= (TF ♯,F♯, TF , ϑLF)

is a special symplectic manifold — theLagrangian special symplectic manifold.


In summary, the magnetized Tulczyjew triple (5.10) yields two special symplectic struc-tures onTF ♯, the one with Lagrangian Liouville one-formϑL

Fis called theLagrangian

special symplectic structureand the one with Hamiltonian Liouville one-formϑHF

iscalled theHamiltonian special symplectic structure.

6. TULCZYJEW’ S APPROACH FOR PARTICLES IN GAUGE FIELDS

Tulczyjew’s approach to classical mechanics has two sides:the Hamiltonian side andthe Lagrangian side, and the two sides are unified under a common geometric setting: thedynamics of a particle with configuration spaceX is determined by a Lagrangian sub-manifoldD of TT ∗X , whose description by its HamiltonianH : T ∗X → R (resp. itsLagrangianL: TX → R) yields the Hamilton (resp. the Euler-Lagrange) equation.

The discussion of Tulczyjew’s approach shall be divided into two parts: the Hamiltonianformulation and the Lagrangian formulation.

6.1. The Hamiltonian formulation. Part of the reasons that Tulczyjew’s approach worksis the fact thatT ∗X is a symplectic manifold. When we replaceT ∗X by a symplecticmanifold (M,ω), the Hamiltonian side still survives because Proposition 4.1 says that(TM,M, τM , ϑ) is special symplectic manifold in which Tulczyjew has introduced [1,page 249]

Definition 6.1 (Tulczyjew’s Hamiltonian system). A Hamiltonian systemin special sym-plectic manifold

(TM,M, τM , ϑ)

is a Lagrangian submanifoldN of (TM, dϑ) such that conditions for the existence of theunique formγ stated in Proposition 4.3 are satisfied andγ is exact. The submanifoldK := τM (N) is called theHamiltonian constraint, and a functionH on K such thatγ = −dH , is called aHamiltonianof N .

In particular, sinceF ♯ is a special symplectic manifold, so we obtain Tulczyjew’sHamiltonian formulation for particles in gauge fields.

To see how this is related to the ordinary Hamiltonian formulation for the dynamics ofparticles without constraint (i.e.,K = M ), one starts with a parametrized smooth curvec on M . By taking derivative, we get a smooth parametrized curvec′ on TM . Now theHamilton equation forc is nothing but the statement thatc′ is a smooth parametrized curveon the Lagrangian submanifoldN of TM . To verify this we just need to work locally,so we may assume that(M,ω) = (T ∗Rn, ωRn) (Darboux’s theorem), thenc = (q, p), soc′ = (q, p, q′, p′). Since the Lagrangian submanifoldN can be locally described as the setof points

(q, p,

∂H

∂p,−

∂H

∂q

),

soc′ is a a smooth parametrized curve onN means precisely that the Hamilton equation

q′ =∂H

∂p= {q,H}, p′ = −

∂H

∂q= {p,H}(6.1)

is satisfied.

16 GUOWU MENG

6.2. The Lagrangian formulation. In this subsection we shall assume thatG is a com-pact connected Lie group,P

p→ X is a principalG-bundle with a fixed principal con-

nection formΘ, Fρ→ X is the associated fiber bundle with fiberF and the associated

G-connection. We further assume thatF is a hamiltonianG-space with (G-equivariant)moment mapΦ, symplectic formΩ. Recall thatSL

F:= (TF ♯,F♯, TF , ϑ

LF) is a special

symplectic manifold. Mimicking Tulczyjew [1, page 251], wehave the following defini-tion.

Definition 1 (Main Definition). A Lagrangian systemin SLF

is a Lagrangian submanifoldN of (TF ♯, dϑL

F) such that conditions for the existence of the unique formγ stated in

Proposition 4.3 are satisfied andγ is exact. The submanifoldJ := TF(N) is called theLagrangian constraint, and a functionL onJ such thatγ = dL, is called aLagrangianofN .

Note that whenG is trivial andF is a point, we haveSLF

= SLX , then the abovedefinition becomes Tulczyjew’s definition 4.4 in Ref. [1].

To get the Euler-Lagrange equation for the dynamics withoutconstraint (i.e.,J = F♯), one starts with a parametrized smooth curveγ onF , then we get a parametrized smoothcurve(γ, (ρ ◦ γ)′) onF♯, so we arrive at a parametrized smooth curvec onT ∗F definedby the following commutative diagram

R

F F♯ T∗F♯ TF

♯ F ♯.

γ

(γ, (ρ ◦ γ)′)

c

c′

τ̃X dL

LegL

α−1F

∼= τF♯

By taking derivative, we get a smooth parametrized curvec′ on TF ♯. Now the Euler-Lagrange equation forγ is nothing but the statement thatc′ is a smooth parametrizedcurve on the Lagrangian submanifoldN of TF ♯. In terms of local coordinates, we canwritten γ = (q, z). This really means that, locally we representγ by (q, z) ◦ γ, but fornotational sanity,(q, z) ◦ γ is also denoted by(q, z). So (q, z) is either a local functiononF or a local function onR, depending on the context. With this understood, we havec = (q, ∂L

∂q̇|q̇=q′ , z), so

c′ =

(q,

∂L

∂q̇

∣∣∣∣q̇=q′

, z, q′,d

dt

(∂L

∂q̇

∣∣∣∣q̇=q′

), z′

).

Since the Lagrangian submanifoldN can be locally described as the set of points(q,

∂L

∂q̇, z, q̇,

∂L

∂qi+ 〈q̇j(∂jAi − ∂iAj),Φ〉+ 〈Ai, ż

β∂βΦ〉, żα

)

whereżα =(∂βL− 〈q̇

iAi, ∂βΦ〉)Ωβα with [Ωαβ ] = [Ωαβ ]−1. Therefore,c′ is a a smooth

parametrized curve onN means that, locally

dzα

dt=

(∂L

∂zβ

∣∣∣∣q̇=q′

−dqk

dt

〈Ak,

∂Φ

∂zβ

〉)Ωβα

d

dt

(∂L

∂q̇i

∣∣∣∣q̇=q′

)=

∂L

∂qi

∣∣∣∣q̇=q′

+dqj

dt

〈∂Ai

∂qj−

∂Aj

∂qi,Φ

〉+

dzα

dt

〈Ai,

∂Φ

∂zα

〉(6.2)


Theorem 1 (Main Theorem). Assume the data in the beginning of this subsection andL:F♯ → R is a smooth map. For the unconstrained particle dynamics with configurationspaceX , internal spaceF , gauge fieldΘ, and LagrangianL, the following statements aretrue.

(1) The equation of motion can be locally written as Eq.(6.2).(2) If (x, y) is a Darboux (local) coordinate for(F,Ω) so thatΩ = dx ∧ dy locally,

then Eq.(6.2) is the Euler-Lagrange equation for Lagrangian

L := L− 〈q̇iAi,Φ〉+ x · ẏ.(6.3)

(3) The action functional for homologically trivial smooth loop γ: S1 → F is

S[γ] :=

∫

S1(γ, (ρ ◦ γ)′)∗L dt+

∫

Σ

γ̃∗ΩΘ(6.4)

whereΣ is an oriented compact2-manifold withS1 = ∂Σ, γ̃: Σ → F is a smoothextension ofγ, andΩΘ is the Sternberg form onF .

(4) There is a generalized version of Dirac’s charge quantization condition:[ΩΘ2π

]∈ Range

(H2(F ;Z)

⊗Z1−→ H2(F ;R)).(6.5)

I.e., the cohomology class represented by the closed real differential two-formΩΘ2πis an integral lattice point of the 2nd cohomology group ofF with real coefficient.

Proof. LagrangianL in Eq. (6.3) emerges out of a short straightforward computation incalculus of variations. ActionS in Eq. (6.4) is obtained fromL with the help of Stokes’theorem in calculus and the fact that in dimension one the oriented bordism group coin-cides with the integral homology group. To arrive at the charge quantization condition(6.5) we demand the uniqueness ofexp[iS[γ]], and use the fact that in dimension two theoriented bordism group coincides with the integral homology group as well as the universalcoefficient theorem in algebraic topology. �

It is equally easy to write down the equation of motion locally for a Lagrangian systemwith a generic Lagrangian constraint.

Remark1. The generalized Dirac quantization condition (6.5) is equivalent to the conditionthat the Sternberg phase is prequantizable, i.e.,[ωΘ] ∈ H2(F ♯;R) is integral. That isbecauseF ♯ andF are homotopy equivalent, andωX is an exact differential 2-form.

Remark2. The main theorem in particular tells us how to incorporate a Yang-Mills fieldin the Lagrangian formulation: just shift the ordinary LagrangianL(q, q̇) by two terms:the Lorentz term−〈q̇iAi(q),Φ(y, x)〉 and the new extra termx · ẏ. Note thatx · ẏ can bereplaced by−y · ẋ or 12 (x · ẏ − ẏ · x), etc.

For simplicity we shall writeddt

(∂L∂q̇i

∣∣∣q̇=q′

)as d

dt

(∂L∂q̇i

), ∂L

∂qi

∣∣∣q̇=q′

as ∂L∂qi

. Let

∂FL :=∂L∂zα

dzα, {f, g}F := Ωαβ ∂f

∂zα∂g∂zβ

Dzdt

:= dzα

dt∂

∂zα+ Yq′·A, Fji :=

∂Ai∂qj

−∂Aj∂qi

+ [Ai, Aj ].

With the help of Eqs. (2.3) and (2.4), we have

18 GUOWU MENG

Corollary 1. Equation(6.2)becomes

d

dt

(∂L

∂q̇i

)=

∂L

∂qi+

dqj

dt〈Fji,Φ〉+ {L, 〈Ai,Φ〉}F

Dz

dtyΩ = ∂FL

(6.6)

provided thatF is a homogeneousHamiltonianG-space.

Example 1 (Dynamics of electrically charged particles). WhenG = U(1), F is a co-adjoint orbit ofG (hence a point−qe ∈ R) with Φ being the inclusion map, the secondequation becomes0 = 0, so the equation of motion becomes the more familiar equation

d

dt

(∂L

∂q̇

)=

∂L

∂q− qe q

′yF.(6.7)

In particular, ifX = R3, L = 12 ṙ · ṙ andB = F23 i + F31 j + F12 k, Eq. (6.7) becomesthe textbook equation of motion

r′′ = qe r′ ×B

with m = 1 andc = 1. In caseX = R3 \ {0}, andB = qm rr3 , condition (6.5) becomesDirac’scharge quantization condition[15]:

qe qm ∈1

2Z

with ~ = 1 andc = 1.

Example 2 (Wong’s equations [3]). With the data in this subsection, we further assumethatX is a Lorentzian manifold with Lorentz metricg, L = 12gµν q̇

µq̇ν , F is a co-adjoint

orbit of G andΦ is the inclusion map. Letpµ = gµν q̇ν , Φ = −ξaT̂ a whereT̂ a’s form abasis forg∗, then the two equations in Eq. (6.6) become Wong’s equations[5, equations(1a) and (1b)] constrained to the Sternberg phase spaceF ♯. SinceF ♯ is a symplectic leafof Wong’s phase space (a Poisson manifold) and solutions to Wong’s equations are alwaysconstrained to symplectic leaves of Wong’s phase space [5, Theorem 2], we effectivelyarrive at Wong’s equations on Wong’s phase space, for which aLagrangian was constructedin Ref. [19, equation (2.6)].

Example 3 (Magnetized Kepler problems [21]). Let k ≥ 1 be an integer andµ be a realnumber, we consider the(2k+1)-dimensional magnetized Kepler problem with magneticchargeµ. Here,X = R2k+1 \ {0}, G = SO(2k), P → X is the pullback of the principalG-bundle ofSO(2k + 1) → S2k under mapr 7→ r|r| , Θ is the pullback of the canonical

invariant connection onSO(2k + 1) → S2k, F is the co-adjoint orbitOµ of G andΦ isjust the inclusion map; see Ref. [21] for more details. The Hamiltonian is

H =1

2p · p−

1

r+

µ2/2k

r2,

the pullback of a real function onT ∗X , and the Lagrangian is

L =1

2ṙ · ṙ+

1

r−

µ2/2k

r2,

the pullback of a real function onTX . It is clear thatH andL are related by the Legendretransformation. Ifk > 1, the generalized Dirac quantization condition (6.5) is equivalentto the condition that the co-adjoint orbitOµ is prequantizable.


Remark3. In the geometric approach here the Lagrangian takes simplerform L and theequation of motion takes complicated form. In the textbook’s approach to dynamics ofelectrically charged particles, the Lagrangian takes complicated formL and the equationof motion is still the simple-looking Euler-Lagrange equation. A similar remark is validfor the Hamiltonian approaches. ( Sternberg’s Hamiltonianapproach is our geometric ap-proach to unconstrained Hamiltonian systems.) As usual, the geometric approach capturesthe essence of the problem, hence works much more generally.

Remark4. The dynamics of a charged particle is a Lagrangian submanifold of the symplec-tic manifoldTF ♯, and the two distinct special symplectic structures onTF ♯ correspondto the two formulations of dynamics, one is Hamiltonian and one is Lagrangian. Sincea Lagrangian submanifold of the symplectic manifoldTF ♯ may not come from a realfunction on eitherF ♯ or F♯, this geometric approach goes much beyond what Sternberg’sHamiltonian approach can handle.

Let us conclude this subsection with the following diagram:

T ∗NF♯ T ∗F ♯ TF ♯ T ∗F♯ T

∗JF♯

T ∗N N F ♯ F♯ J T∗J

R F R

R

πF♯

βF αF

τF♯ TF πF♯

πN

H−dH

π̃∗ π̃ LdL

πJ

γ

(γ, (ρ ◦ γ)′)

6.3. The Legendre transformation. For completeness we give a sketch of the Legendretransformation for the experts. The general readers who wish to know more details shouldconsult sections 5 and 6 in Ref. [1].

Definition 6.2 (The Legendre transformation). The identity symplectic diffeomorphism1TF♯ from S

LF

to SHF

is called the Legendre transformation, and the identity symplecticdiffeomorphism1TF♯ fromS

HF

to SLF

is called the inverse Legendre transformation.

SinceϑLF

− ϑHF

= dϑ̂X , we have

Proposition 6.1. The Legendre transformation is generated by−ΦF where

ΦF : F♯ ×F F♯ → R

is the map that sends((q, q̇, z), (q, p, z)) to 〈q̇, p〉. The inverse Legendre transformation isgenerated by

Φ̃F : F♯ ×F F♯ → R

is the map that sends((q, p, z), (q, q̇, z)) to 〈q̇, p〉.

If f is a function ofx with a unique critical pointx0, we useStatx[f(x)] to denotef(x0).

Example 4. Let N be a Lagrangian submanifold ofTF ♯ which is a generated by a mapL: F♯ → R and also a map−H : F ♯ → R. ThenL andH are related byH(y) =Statx[Φ(x, y)−L(x)] subject to constraint̃τX(x) = π̃X(y), andL(x) = Staty[Φ̃(y, x)−H(x)] subject to the same constraint. In terms of local coordinates, we haveH(q, p, z) =

20 GUOWU MENG

p · q̇ − L(q, q̇, z) evaluated aṫq such that∂L∂q̇i

= pi, andL(q, q̇, z) = p · q̇ − H(q, p, z)

evaluated atp such that∂H∂pi

= qi.

APPENDIX A. USEFUL FACTS

A.1. Tangent lift [18]. Let X be a smooth manifold andTX be its total tangent space.The tangent lift, denoted bydT , is a map the sends a differentialk-form onX to a differ-entialk-form onTX . By definition,

dT = d ◦ iT + iT ◦ d(A.1)

whereiT is a map that sends a differential(k + 1)-form onX to a differentialk-form onTX as follows: forα ∈ Ωk+1(X), iT (α) ∈ Ωk(TX) is defined via equation

iT (α)|v = ṽy τ∗Xα|v for anyv ∈ TX.(A.2)

Hereṽ ∈ TvTX is any horizontal lift ofv ∈ TτX(v)X . For simplicity, we use the samesymbol for a differential form onX and its pullback onTX . Then, in terms of localcoordinateq and(q, q̇) onX andTX , we have

iT(αi0···ik dq

i0 ∧ · · · ∧ dqik)=

k∑

j=0

(−1)j q̇ijαi0···ik dqi0 ∧ · · · d̂qij · · · ∧ dqik

where the hat ondqij means thatdqij is missing.

A.2. “Cotangent map” T ∗f . Let f : Y → X be a smooth map. While there is a canonicalmorphism from the tangent bundle ofY to the tangent bundle ofX , there is no canonicalmorphism from the cotangent bundle ofY to the cotangent bundle ofX . However, iff :Y → X is a fiber bundle with a (Ehresmann) connection, then, for anyy ∈ Y , if Hyis the horizontal subspace ofTyY , we haveTf(y)X ∼= Hy ⊂ TyY , so we have a linearmapT ∗y Y → T

∗f(y)X which, upon being globalized, becomes the top arrowT

∗f in the

commutative square

T ∗Y T ∗X

Y X

T∗f

πY πX

f

which is a morphism from the cotangent bundle ofY to the cotangent bundle ofX .

A.2.1. Local formula forT ∗f . Here we assume thatf : Y → X is the fiber bundle with fiber

F , associated to the principalG-bundlePp→ X with a principal connection, sof : Y → X

has an associatedG-connection. To work out a local formula forT ∗f : T∗Y → T ∗X , we

may assume thatX is diffeomorphic toRn, Pp→ X is trivial, andF is diffeomorphic to

Rl. Upon fixing a diffeomorphismq: X → Rn, a diffeomorphismz: F → Rl, and a local

trivialization

P X ×G

X

p p1

φ

∼=


we have a diffeomorphismY ∼= Rn × Rl which shall be denoted by(q, z). We also havediffeomorphisms

T ∗Y ∼= Rn × Rl × (Rn × Rl)∗ ∼= (Rn × Rl)2 and T ∗X ∼= Rn × (Rn)∗ ∼= (Rn)2

and they shall be denoted by(q, z, p, y) and(q, p) respectively. Finally, the infinitesimalaction ofg onF assigns a vector fieldYξ onF to eachξ ∈ g. Let us also useYξ to denotethe coordinator vector ofYξ with respect to the local tangent frame∂∂zα .

Under the trivializationφ, the principal connection onPp→ X is represented by ag-

valued differential one-formA · dq := Ai dqi onX , so the horizontal tangent vector ofYcan be represented by

(q, z, q̇,−Yq̇·A).

Therefore, the mapT ∗YT∗f→ T ∗X can be represented by

(q, z, p, y) 7→(qi, pj − y · YAj

).(A.3)

22 GUOWU MENG

APPENDIX B. A LIST OF SYMBOLS

y the interior product of vectors with forms∧ the wedge product of formsd the exterior derivative operatordT the tangent lift operatorτX : TX → X the tangent bundle projectionπX : T

∗X → X the cotangent bundle projectionϑX , ωX := dϑX the Liouville form, the symplectic form onT ∗Xϑ̂X : TT

∗X → R the map such that̂ϑX |Tα(T∗X) = ϑX |αf∗ the pullback on differential forms under smooth mapfTf the tangent map offTmf the linearization off at pointmT ∗f see appendix A for definitionq, (q, q̇), (q, p) local coordinate map onX , TX , T ∗Xπ : E → X a real vector bundle overXpE : T

∗E → E∗ see the explanation right before diagram (3.2)(q, u), (q, α), (q, û) local coordinate map onE, E∗, E∗∗

q(t) a smooth parametrized curve onXq′(t) the derivative ofq(t), it is a smooth parametrized curve onTXι : E → E∗∗ the usual identificationG a compact connected Lie group, viewed as a

Lie subgroup of a rotation groupg, g∗ the Lie algebra ofG and its dualξ an element ing〈 , 〉 the paring of elements in vector spaceV with elements inV ∗

Ada the adjoint action ofa ∈ G ongP → X a principalG-bundleΘ ag-valued differential one-form onP that

defines a principal connection onP → XRa the right action onP by a ∈ GXξ the vector field onP which represents the

infinitesimal right action onP by ξ ∈ gφ : X ×G → P a local trivialization ofP → XAφ or simplyA ag-valued differential one-form onX which

locally representsΘ under trivializationφF a hamiltonianG-spaceF := P ×G F P × F quotient by equivalence relation(p, f) ∼ (Ra−1(p), La(f))z, (z, y) local coordinate map onF , T ∗FΦ : F → g∗ theG-equivariant moment mapLa the left action onF by a ∈ GYξ the vector field onF which represents the infinitesimal left action

onF by ξ ∈ g. It also denotes the coordinate vector ofYξF

ρ→ X the associated fiber bundle with fiberF

F ♯ρ♯

→ T ∗X,F♯ρ♯→ TX the pullback bundle ofF

ρ→ X underπX , τX

one formsϑLF

andϑHF

see Subsection 5.2.2 for their definition


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DEPARTMENT OF MATHEMATICS, HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY, CLEARWATER BAY, KOWLOON, HONG KONG

E-mail address: [email protected]

1. Introduction2. Sternberg phase space2.1. Principal connection2.2. Sternberg form on F2.3. Sternberg phase space

3. A canonical isomorphism of double vector bundles3.1. Local formulae

4. Special symplectic manifold5. Tulczyjew triple and its magnetized version5.1. Classical Tulczyjew triple5.2. Magnetized Tulczyjew triple

6. Tulczyjew's approach for particles in gauge fields6.1. The Hamiltonian formulation6.2. The Lagrangian formulation6.3. The Legendre transformation

Appendix A. Useful factsA.1. Tangent lift pidello1987derivationA.2. ``Cotangent map" T*f

Appendix B. A list of symbolsReferences

Documents

arXiv:1405.0748v2 [math-ph] 27 May 2014arXiv:1405.0748v2 [math-ph] 27 May 2014 TULCZYJEW’S APPROACH FOR PARTICLES IN GAUGE FIELDS GUOWU MENG ABSTRACT.In mid-1970s W. M. Tulczyjew