lagrangian submanifolds and the hamilton–jacobi equation

Monatsh Math (2013) 171:269–290DOI 10.1007/s00605-013-0522-1

Lagrangian submanifolds and the Hamilton–Jacobiequation

María Barbero-Liñán · Manuel de León ·David Martín de Diego

Received: 5 September 2012 / Accepted: 18 May 2013 / Published online: 2 June 2013© Springer-Verlag Wien 2013

Abstract Lagrangian submanifolds are becoming a very essential tool to general-ize and geometrically understand results and procedures in the area of mathemat-ical physics. The geometric version of the Hamilton–Jacobi equation in terms ofLagrangian submanifolds enables here some novel interesting applications of theHamilton–Jacobi equation in holonomic, nonholonomic and time-dependent dynam-ics from a geometrical point of view.

Keywords Lagrangian submanifolds · Hamilton–Jacobi equation ·Holonomic and nonholonomic Hamilton–Jacobi equation ·Time-dependent Hamilton–Jacobi equation

Mathematics Subject Classification (2010) 37J60 · 53D05 · 53D12 · 70H20

Communicated by A. Cap.

M. Barbero-Liñán (B)Departamento de Matemáticas, Universidad Carlos III de Madrid,Avenida de la Universidad 30, 28911 Leganés, Madrid, Spaine-mail: [email protected]

M. Barbero-Liñán · M. de León · D. M. de DiegoInstituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM),C/Nicolás Cabrera 13-15, 28049 Madrid, Spaine-mail: [email protected]

D. M. de Diegoe-mail: [email protected]

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270 M. Barbero-Liñán et al.

1 Introduction

Hamilton–Jacobi theory is useful to study the solution of partial differential equationsby analyzing the solutions of a system of ordinary differential equation and viceversa. Roughly speaking, the autonomous Hamilton–Jacobi problem for a Hamiltoniansystem on a configuration manifold Q consists of finding a function S : Q → R, q →S(q), such that

H

(q,∂S

∂q

)= E,

where H : T ∗Q → R is the Hamiltonian function defined on the cotangent bundleT ∗Q and E is a constant. Hamilton–Jacobi theory is particularly relevant in classicalmechanics and calculus of variations (see, for instance [13,17,24] for some applica-tions).

Recently, a new geometric approach to Hamilton–Jacobi theory has been devel-oped [6,8–10,26]. As a result, new developments of Hamilton–Jacobi theory for non-holonomic mechanical systems [11,16,19] and in the discrete setting [28] have beenobtained. On the other hand, the notion of Lagrangian submanifolds, which are distin-guished submanifolds of a symplectic manifold [32], has gained a lot of interest dueto its applications on dynamics [18,30,31]. These applications do not only appear onclassical dynamics, but also on more algebraic structures that generalize the notion oftangent bundle: the so-called Lie algebroids [20] and Lie affgebroids [15].

In this paper we review how general Lagrangian submanifolds are used to extendthe geometric version of classical Hamilton–Jacobi equation along similar lines asin [6,8,9,26], but bringing together the Lagrangian and Hamiltonian formalism. In[1,23], Lagrangian submanifolds obtained from a one-closed form allowed to reinter-pret the classical Hamilton–Jacobi theorem in a more geometric way. More generalLagrangian submanifolds, such as the ones generated by Morse families, are usedin the literature to describe a generalized version of the Hamilton–Jacobi equation(see [6,26]). Here, we consider a particular kind of Lagrangian submanifolds definedin Lemma 4.2, which allows us to describe the Hamilton–Jacobi equation for holo-nomic and nonholonomic dynamics. Nowadays, nonholonomic constraints are one ofthe most fascinating and applied topics in mechanics. There are many open questionsrelated with this subject: characterization of the integrability, construction of geometricintegrators, stabilization, controllability, etc. Moreover, systems with nonholonomicconstraints appear in applied fields such as engineering and robotics (see [7,12] andreferences therein). That is why the characterization of the Hamilton–Jacobi equationfor these systems might help to elucidate many of the above-mentioned open questions.On the other hand, we are also able to describe the time-dependent Hamilton–Jacobiequation within the same geometric framework by means of a family of Lagrangiansubmanifolds of T ∗Q. The main contribution of this paper is the application of onlyLagrangian submanifold theory to obtain a geometric version of Hamilton–Jacobiequation in both holonomic and nonholonomic dynamics.

The paper is organized as follows. In Sect. 2, we first review the notion of Lagrangiansubmanifolds and its characterizations. Section 3 recalls the classical Hamilton–Jacobi

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Lagrangian submanifolds and the Hamilton–Jacobi equation 271

theory for both autonomous and non-autonomous cases. Section 4 contains the geomet-ric interpretation of autonomous Hamilton–Jacobi equation for a general connectedLagrangian submanifold in both Lagrangian and Hamiltonian formalism. The defini-tion of a particular kind of Lagrangian submanifold allows us to give a geometric inter-pretation of Hamilton–Jacobi equation which is useful to describe Hamilton–Jacobiequation for holonomic and nonholonomic dynamics in Sect. 5. For the nonholonomiccase we need the language of Lagrangian distributions on symplectic vector bundlesto obtain the Hamilton–Jacobi equation, see Proposition 5.2. Another interesting con-tribution of this paper is the use of a family of Lagrangian submanifolds to give ageneralized version of the non-autonomous Hamilton–Jacobi equation in Sect. 6. As aresult, this equation is described for holonomic and nonholonomic dynamics in Sect. 7.

2 Lagrangian submanifolds

The essential background on symplectic geometry is reviewed here to make the paperas much self-contained as possible. Lagrangian submanifolds [32] are the extensionto manifolds of the notion of Lagrangian subspaces of symplectic vector spaces.

Let us recall that a symplectic vector space is a pair (E,�) where E is a vectorspace and � : E × E → R is a skew-symmetric bilinear map of maximal rank. See[14,22,23,32] for more details.

Definition 2.1 Let (E,�) be a symplectic vector space and F ⊂ E a subspace. The�-orthogonal complement of F is the subspace defined by

F⊥ = {e ∈ E | �(e, e′) = 0 for all e′ ∈ F}.

The subspace F is said to be

(i) isotropic if F ⊆ F⊥, that is, �(e, e′) = 0 for all e, e′ ∈ F .(ii) Lagrangian if F is isotropic and has an isotropic complement, that is, E = F⊕F ′,

where F ′ is isotropic.

A well-known characterization of Lagrangian subspaces of finite dimensional sym-plectic vector spaces is summarized in the following result:

Proposition 2.2 Let (E,�) be a finite dimensional symplectic vector space and F ⊂E a subspace. Then the following assertions are equivalent:

(i) F is Lagrangian,(ii) F = F⊥,

(iii) F is isotropic and dim F = 12 dim E.

As a consequence, we can characterize a Lagrangian subspace by checking if it hashalf the dimension of E and �|F = 0.

Remember that a symplectic manifold (M, ω) is defined by a differentiable man-ifold M and a non-degenerate closed 2-form ω on M . Therefore, for each x ∈M, (Tx M, ωx ) is a symplectic vector space and a symplectic manifold has evendimension.

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A symplectic vector bundle (E, ω, π) is a vector bundle π : E → M over a man-ifold M equipped with a smooth field ω of fiberwise nondegenerate skew-symmetricbilinear maps ωx : Ex × Ex → R. Therefore, if (M, ω) is a symplectic manifold, then(T M, ω, τM ) is a symplectic vector bundle, where τM : T M → M is the canonicaltangent projection. The converse is only true ifω is closed. Let D be a vector subbundleof a symplectic vector bundle (E, ω, π). We say that D is a Lagrangian vector subbun-dle if every fiber Dx is a Lagrange subspace of (Ex , ωx ). In the particular case, when(M, ω) is a symplectic manifold, a Lagrangian vector subbundle D of (T M, ω, τM )

will be called a Lagrangian distribution.The notion of Lagrangian subspace can be transferred to submanifolds by requesting

that the tangent space of the submanifold is a Lagrangian subspace for every point inthe submanifold of a symplectic manifold.

Definition 2.3 Let (M, ω) be a symplectic manifold and i : N → M an immersion.It is said that N is an isotropic immersed submanifold of (M, ω) if (Tx i)(Tx N ) ⊂Ti(x)M is an isotropic subspace for each x ∈ N . A submanifold N ⊂ M is calledLagrangian if it is isotropic and there is an isotropic subbundle P ⊂ T M |N such thatT M |N = T N ⊕ P .

Note that i : N → M is isotropic if and only if i∗ω=0, that is,ω(Tx i(vx ),Tx i(ux ))=0 for every ux , vx ∈Tx N and for every x ∈ N .

Of course, all the previous notions can be easily extended to the case of symplecticvector bundles. These extensions will be used in the sequel.

The canonical model of symplectic manifold is the cotangent bundle T ∗Q of anarbitrary manifold Q. Denote by πQ : T ∗Q → Q the canonical projection and definea canonical 1-form θQ on T ∗Q by

(θQ

)αq(Xαq ) = 〈αq ,TαqπQ(Xαq )〉,

where Xαq ∈ Tαq T ∗Q, αq ∈ T ∗Q and q ∈ Q. If we consider bundle coordinates(qi , pi ) on T ∗Q such that πQ(qi , pi ) = qi , then

θQ = pi dqi .

The 2-form ωQ = −dθQ is a symplectic form on T ∗Q with local expression

ωQ = dqi ∧ d pi .

The Darboux theorem states that this is the local model for an arbitrary symplecticmanifold (M, ω): there exist local coordinates (qi , pi ) in a neighbourhood of eachpoint in M such that ω = dqi ∧ d pi .

Note that the canonical 1-form θQ is universal in the sense that γ ∗(θQ) = γ for anarbitrary 1-form γ on Q. Hence γ ∗(ωQ) = −dγ .

A relevant example of Lagrangian submanifold of the cotangent bundle is the fol-lowing one.

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Proposition 2.4 [23] Let γ be a 1-form on Q and L = Im γ ⊂ T ∗Q. The submanifoldL of T ∗Q is Lagrangian if and only if γ is closed.

The result follows because dim L = dim Q and γ ∗(ωQ) = −dγ .Given a symplectic manifold (P, ω), dim P = 2n, it is well-known that its tangent

bundle T P is equipped with a symplectic structure denoted by dTω, where dTω

denotes the tangent lift of ω to T P . If we take Darboux coordinates (qi , pi ) on P ,that is, ω = dqi ∧ d pi , then dTω = dqi ∧ d pi + dqi ∧ d pi , where (qi , pi ; qi , pi ) arethe induced coordinates on T P . We will denote the bundle coordinates on T ∗ P by(qi , pi ; ai , bi ), then ωP = dqi ∧dai +d pi ∧dbi . If we denote by : T P → T ∗ P theisomorphism defined by ω, that is, ω(v) = iv ω, then we have ω(qi , pi ; qi , pi ) =(qi , pi ;− pi , qi ). This isomorphism plays an important role on the description of thedynamics of Lagrangian and Hamiltonian systems (more details can be found in [30]).

Given a function H : P → R, and its associated Hamiltonian vector field X H ,that is, iX Hω = dH , then the image of X H , Im X H , is a Lagrangian submanifold of(T P, dTω).

In Sects. 4 and 6 we will show other examples of Lagrangian submanifolds (seealso [23]) that play a key role in developing the novel applications of Lagrangiansubmanifolds on Hamilton–Jacobi theory described in this paper.

3 Classical Hamilton–Jacobi theory

Once we have introduced the necessary background on symplectic geometry, let usreview briefly Hamiltonian dynamics. If H : T ∗Q → R is a Hamiltonian function,the Hamiltonian vector field X H on T ∗Q is the solution of the equation

iX HωQ = dH.

If we take Darboux coordinates (qi , pi ) on T ∗Q, Hamilton’s equations are locallywritten as

dqi

dt= ∂H

∂pi(q, p),

d pi

dt= −∂H

∂qi(q, p).

As described in [1,2], solutions of the Hamilton–Jacobi equation may help with inte-grating Hamilton’s equations and obtaining qualitative information about the dynam-ics. Let us review here the classical Hamilton–Jacobi theory both in the autonomousand non-autonomous cases.

The following is a geometric version of the classical Hamilton–Jacobi theory [1,2]in the autonomous case.

Theorem 3.1 (Hamilton–Jacobi equation [1]) Let T ∗Q be the symplectic manifoldwith the symplectic structure ωQ = −dθQ. Let X H be a Hamiltonian vector field on

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T ∗Q, and let S : Q → R be a smooth function. The following two conditions areequivalent:

(i) for every curve c in Q satisfying

c′(t) = TπQ(X H (dS(c(t))))

the curve t → dS(c(t)) is an integral curve of X H .(ii) S satisfies the Hamilton–Jacobi equation H ◦ dS = E, a constant, that is,

H

(q,∂S

∂q

)= E .

Remark 3.2 The above theorem describes solutions to Hamilton–Jacobi equation byfinding particular integral curves of the Hamiltonian vector field. Those integral curvesare obtained, when possible, from integral curves of a vector field on Q defined byXdS = TπQ ◦ X H ◦ dS. In other words, if there exists a function S on Q such that thefollowing diagram commutes

T ∗Q

πQ

��

X H �� T T ∗Q

TπQ��

Q

dS

��

XdS�� T Q

then the integral curves c of XdS define the integral curves dS ◦ c of the Hamiltonianvector field X H .

Moreover, note that the set N = {(q, dS(q)) | q ∈ Q} of T ∗Q is a Lagrangiansubmanifold according to Proposition 2.4 since it is defined by a closed 1-form,dS : Q → T ∗Q. Thus the notion of Lagrangian submanifolds already appears inthe classical version of Hamilton–Jacobi equation. Then it makes sense to look foran extension of this result to any Lagrangian submanifold as shown in Sect. 4 andconsider the applications, for instance, to holonomic and nonholonomic dynamics,see Sect. 5.

The classical statement of Hamiton–Jacobi equation for non-autonomous case isthe following:

Theorem 3.3 (Time-dependent Hamilton–Jacobi equation [1]) Let T ∗Q be the sym-plectic manifold with the symplectic structure ωQ = −dθQ. Let X H be a Hamiltonianvector field on T ∗Q and W : R × Q → R be a smooth function. The following twoconditions are equivalent:

(i) for every curve c in Q satisfying

c′(t) = TπQ(X H (dWt (c(t))))

the curve t → dWt (c(t)) is an integral curve of X H , where Wt : Q →R, Wt (q) = W (t, q).

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(ii) W satisfies the Hamilton–Jacobi equation

H(q,∂W

∂q)+ ∂W

∂t= constant on T ∗Q,

that is, H ◦ dWt + ∂W

∂t= K (t).

Remark 3.4 The analogous diagram to the one in Remark 3.2 for the time-dependentcase is the following one:

T ∗Q

πQ

��

X H �� T T ∗Q

TπQ��

Q

dWt

��

XdWt �� T Q

which must be understood as in Remark 3.2 for every time t .

In the non-autonomous case the set Nt = {(q, dWt (q)) | q ∈ Q} is also aLagrangian submanifold of T ∗Q for every t . In Sect. 6 we will discuss the time-dependent Hamilton–Jacobi theorem in an intrinsic way for a general family ofLagrangian submanifolds and a suitable family of submanifolds in T T ∗Q so thateverything comes together.

4 A geometric version of Hamilton–Jacobi equation

According to the philosophy of this paper we use arbitrary Lagrangian submanifolds Lof the cotangent bundle T ∗Q to write geometrically Theorem 3.1, see a more generaldescription in [6,26]. For our purposes we consider a Lagrangian submanifold L inT ∗Q that does not necessarily project over the entire manifold Q as in the case ofTheorem 3.1, see Remark 3.2. Let iL : L → T ∗Q be the canonical immersion, thecanonical tangent lift of iL is denoted by TiL : T L → T T ∗Q.

It is also possible to define the canonical lift of iL to the cotangent bundle, but onlyon the points pq ∈ L. Let 〈·, ·〉 be the inner product, then the canonical lift of iL tothe cotangent bundle, T∗iL : T ∗

LT ∗Q → T ∗L, is defined as follows

〈T∗iL(pq )

iL(�pq ), X pq 〉 = 〈�pq ,Tpq iL(X pq

)〉, (1)

for pq ∈ L, �pq ∈ T ∗LT ∗Q, X pq ∈ Tpq L. This notion corresponds to the pull-back

of iL induced on differential forms.

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The situation can be summarized in the following diagram:

T L � � TiL ��

τL

�� T T ∗Q

τT ∗ Q

��

�� T ∗T ∗Q

πT ∗ Q

��

� � T∗iL �� T ∗L

πL

��

L

X H |L

L

dH |L

Here, the map is the symplectic musical isomorphism with inverse given by� : T ∗T ∗Q → T T ∗Q. Moreover note that in the above diagram we have identifiedL with its image by iL. Now we can generalize Theorem 3.1 to any Lagrangiansubmanifold of T ∗Q (see also [6]).

Theorem 4.1 Let L be a Lagrangian submanifold of T ∗Q. The following statementsare equivalent:

(i) T∗iL (dH |L) = 0, where iL : L ↪→ T ∗Q is the canonical immersion;(ii) Im((dH)|L) ⊂ (T L)0, where (T L)0 is the annihilator of T L;

(iii) Im((X H )|L) ⊂ T L.

Proof Note that condition (i), T∗iL (dH |L) = 0, is equivalent to H |L being constantand it is directly equivalent to condition (i i). Now, let � : T ∗T ∗Q → T T ∗Q be thesymplectic musical isomorphism, then the equivalence between (i i) and (i i i) is basedon the following equivalence:

�((dH)|L) ⊂ �((T L)0) ⇔ X H (pq) ⊂ (Tpq L)⊥ = Tpq L ∀ pq ∈ L,

where we first have used that (TL)⊥ = �((TL)0) by [23, Proposition 13.2]. Observethat we have only used the Lagrangian character of L in the last step because(Tpq L)⊥ = Tpq L for every pq ∈ L. ��

Let us rewrite the above theorem in local coordinates. If L is locally given by theindependent constraints �i (q, p) = 0 for i = 1, . . . , dim Q, then the condition ofbeing Lagrangian is written as

{�i ,� j } = ∂�i

∂qk

∂� j

∂pk− ∂�i

∂pk

∂� j

∂qk= 0, ∀ i, j = 1, . . . , dimQ .

where { · , · } is the canonical Poisson bracket of functions on T ∗Q induced by ωQ .The tangent bundle of L is described locally as follows:

T L ={(q, p, q, p) ∈ T T ∗Q

∣∣∣�i (q, p) = 0,∂�i

∂q jq j + ∂�i

∂p jp j = 0

}

={(q, p, q, p) ∈ T T ∗Q

∣∣∣�i (q, p) = 0, p j = −λk∂�k

∂q j, q j = λk

∂�k

∂p j

},

where λk are functions on L.

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Moreover,

(T L)0 ={(q, p, α, β) ∈ T ∗T ∗Q

∣∣∣�i (q, p) = 0, αi = λk∂�k

∂qi, βi = λk

∂�k

∂pi

}.

The standard case for Lagrangian submanifolds is given by a closed 1-form γ

on Q as mentioned in Proposition 2.4. Observe that Theorem 4.1 is an extension ofTheorem 3.1 as mentioned in Remark 3.2 considering L = Im dS, where S : Q → R.Let us define a vector field XdS

H on Q by

XdSH = TπQ ◦ X H ◦ dS.

Since dS is a diffeomorphism onto its image L = Im dS, it is equivalent to find theintegral curves of XdS

H and to find the integral curves of the vector field X H |L ∈ X(L),see [10].

Another interesting example of Lagrangian submanifolds is given by a submanifoldM immersed in Q, iM : M ↪→ Q, and a closed 1-form γ on M , as follows

LM,γ = {μ ∈ T ∗Q | T∗iM(μ) = γ }= {μ ∈ T ∗Q | 〈μiM(x),Tx iM(vx )〉 = 〈γx , vx 〉 ∀ vx ∈ TxM}, (2)

where T∗iM has been defined in (1).Note that locally, γ = d f where f ∈ C∞(M).

Lemma 4.2 Let γ be a closed 1-form on M. The submanifold LM,γ = {μ ∈T ∗Q | T∗iM(μ) = γ } is a Lagrangian submanifold of T ∗Q.

Proof It can be proved straightforward by using local coordinates and the closednessof γ . Assume that locally iM(qa) = (qa, ψα(qa)), then

LM,γ ={(qa, qα, μa, μα) ∈ T ∗Q | qα = ψα(qa), μa = γa − ∂�α

∂qaμα

}.

Obviously dim LM,γ = 1

2dim T ∗Q. It remains to prove that T∗iLM,γ

(ωQ) = 0,

that is,

ωQ(TiLM,γ(Xμq ),TiLM,γ

(Yμq )) = 0 for all Xμq , Yμq ∈ Tμq LM,γ .

��Remark 4.3 We can introduce the notion of complete solution of the Hamilton–Jacobiequation using also Theorem 4.1. In this case, instead of a fixed Lagrangian subman-ifold we would consider a Lagrangian foliation of T ∗Q, that is, a foliation of T ∗Qwhose leaves are Lagrangian submanifold such that they satisfy the conditions ofTheorem 4.1. In other words, the Hamiltonian vector field X H is tangent to the leavesof the Lagrangian foliation.

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5 Applications of time-independent Hamilton–Jacobi theory

In this section we use those Lagrangian submanifolds defined in (2) to obtain someinteresting and novel applications of Theorem 4.1 in Hamilton–Jacobi theory for theautonomous case (see [6,10]).

5.1 Holonomic dynamics

Let N be a submanifold of Q and h be a Hamiltonian function on T ∗N . Take a closed1-form γ on N . Then, we have that Im γ is a Lagrangian submanifold of T ∗N and atthe same time we can define a Lagrangian submanifold LN ,γ of T ∗Q according to (2)which is given by

LN ,γ = {μ ∈ T ∗Q | T∗iN (μ) = γ }= {μ ∈ T ∗Q | μiN (x)(Tx iN (vx )) = γx (vx ) ∀ vx ∈ Tx N },

where iN : N ↪→ Q is the canonical immersion. The following diagram illustrates theabove situation

R T ∗Nh

πN

�� T ∗Q

πQ∣∣LN ,γ

��

T∗iN LN ,γ� �iLN ,γ

��N N

Hence we can extend the Hamiltonian h to a hamiltonian H : T ∗N Q −→ R just

defining H = h ◦ T∗iN .All the elements in Theorem 4.1 for the Lagrangian submanifold LN ,γ are gathered

in the following diagram:

Im(

X H |LN ,γ

)� �

��Im

(dH |LN ,γ

)�

��

TLN ,γτLN ,γ

��

T iLN ,γ �� TLN ,γ T ∗Q

τT ∗ Q

��

��T ∗LN ,γ

T ∗Q

πT ∗ Q

��

��(TLN ,γ )

0� �

��

LN ,γ

π

��

LN ,γ

π

��N N

Proposition 5.1 The standard Hamilton–Jacobi equation for holonomic dynamicson N

γ ∗dh = 0, that is, h ◦ γ = constant,

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is equivalent to

T∗iLN ,γ (dH) = 0,

where H : T ∗N Q −→ R, H = h ◦ T∗iN .

Proof Let us start from T∗iLN ,γ (dH) = 0 and use the definition of the Hamiltonianfunction H on T ∗

N Q. For μq ∈ LN ,γ ,

0 = T∗iLN ,γ

(dH(μq)

)= T∗iLN ,γ

(d

(h ◦ T∗iN

)(μq)

)= T∗iLN ,γ (d (h ◦ γ (q)))= T∗iLN ,γ

(γ ∗dh(q)

),

where π(μq) = q ∈ N .Since γ ∗dh is πN -basic form, we have

T∗iLN ,γ

(γ ∗dh(q)

) = γ ∗dh(q)

and the result follows. ��

Hence classical Hamilton–Jacobi equation in Theorem 3.1 for a closed 1-formγ on a submanifold N of Q is recovered as the extended Hamilton–Jacobi for theLagrangian submanifold LN ,γ of T ∗Q.

Locally, γ = dS where S is a local function on N and φα are constraints definingimplicitly N . Let S be an arbitrary extension of S to a neighborhood of Q, we havethat

LN ,γ = {μ ∈ T ∗Q | μ = dS + λαdφα}.Therefore, the Hamilton–Jacobi equation T∗iLN ,γ (dH) = 0 is equivalent to

H(dS(q)+ λαdφα(q)) = constant ∀ q ∈ N .

That is, we have the following system of equations

H

(qi ,

∂ S

∂qi+ λα

∂φα

∂qi

)= constant,

φα(qi ) = 0.

⎫⎪⎬⎪⎭

On the other hand, the submanifold N can be locally immersed in Q as followsiN : (qa) ↪→ (qa, �α(qa)). Then the constraints φα become φα(q) = qα − �α(qa)

and T∗iN : T ∗N Q → T ∗N is locally given by

(qa, pi ) →(

qa, pa + pα∂�α

∂qa

).

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From here, we have

H|LN ,γ (qi , pi ) = H(qi , dS(q)+ λαdφα(q))

= H

(qi ,

∂ S

∂qa− λα

∂�α

∂qa, λα

)

= (h ◦ i∗N )(

qi ,∂ S

∂qa− λα

∂�α

∂qa, λα

)

= h

(qa,

∂S

∂qa

).

Thus we obtain the Hamilton–Jacobi equation on N , h ◦ dS = constant, from theHamilton–Jacobi equation on T ∗Q as stated in Proposition 5.1.

5.2 Nonholonomic mechanics

Consider a mechanical system specified by a regular lagrangian L : T Q −→ R and asubmanifold D of T Q called the constraint submanifold. If the submanifold D is notof the form T N for a submanifold N of Q, the constraints are called nonholonomic.We restrict ourselves to the case of linear constraints where D is a vector subbundle ofT Q. The necessary background to state the main results of this section can be foundin detail in [21] (see also [16,19]).

The regular Lagrangian defines a vector field �L on T ∗Q by the equation

i�LωL = dEL ,

where ωL = −d�L is the Poincaré-Cartan 2-form where �L = S∗(dL) and EL =�L − L is the energy of the system. The regularity condition of the Lagrangian L isequivalent to the symplecticity of the 2-form ωL . Here,� is the Liouville vector fieldon T Q and the canonical tensor field S is called the vertical endomorphism. In localnatural coordinates (qi , qi ) on T Q by

� = qi ∂

∂qi, S = ∂

∂qi⊗ dqi .

The vector field�L is a second order vector field (SODE), that is, it verifies S(�L) = �,but it is not in general tangent to D. That is why it is necessary to introduce forces tomodify the dynamics in such a way that the solutions are integral curves of a vectorfield tangent to D. Let us assume that there is a set of reaction forces, mathematicallyexpressed as a set F ⊂ T ∗

DT Q. Therefore, the equations of the dynamics are nowgiven by

i�nhωL − dEL ∈ F, �nh ∈ X(D) , (3)

where X(D) is the space of vector fields on D.

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For nonholonomic systems the reaction forces are of “mechanical type” (see [25])or, in other words, they are given by semibasic 1-forms, that is, S∗(μ) = 0 for all μ ∈F . Therefore, the possible solutions of (3) automatically verify the SODE condition,that is, S(�nh) = �. A special choice for the reaction forces is given by Chetaev’scondition which can be intrinsically given by F = S∗(T D0). Under this conditionthe system is conservative. In other words, the energy EL remains constant along themotion. From now on, we restrict ourselves to this case.

Denote by W ⊂ TDT Q the subbundle defined by W 0 = F . In the sequel, weassume the following compatibility condition

TD(T Q) = T D ⊕ W ⊥ , (4)

where W ⊥ denotes the symplectic orthogonal to W with respect to the symplecticform ωL . Observe that by construction W ⊥ is contained in the vertical tangent bundleto T Q on D.

Condition (4) automatically implies the existence of a unique solution�nh of Eq. (3).Now let us assume that the Lagrangian is hyper-regular, that is, the Legendre trans-formation FL : T Q −→ T ∗Q, locally defined by

FL(qi , qi ) =(

qi , pi = ∂L

∂qi

),

is a diffeomorphism. Under this assumption we have an equivalent hamiltonian for-mulation for the nonholonomic mechanics.Then the new ingredients are a hamiltonianH : T ∗Q → R defined by H = EL ◦ (FL)−1, the constraint submanifold given byC = FL(D) and the vector subbundle F ∈ TC T ∗Q defined by (FL)∗ F = F . We alsohave W = (FL)∗W , then the compatibility condition (4) is rewritten as

TC (T∗Q) = T C ⊕ W ⊥,

where W ⊥ as above denotes the symplectic orthogonal to W with respect to thecanonical symplectic form ωQ . Moreover, W ⊥ is contained in the vertical tangentbundle to T ∗Q. Consequently, W ⊥ is isotropic. Thus,

W = W ∩ (T C ⊕ W ⊥) = (W ∩ T C)⊕ W ⊥,

and we conclude that H = W ∩ T C is a symplectic vector bundle on C , that is,

H ⊕ H⊥ = TC T ∗Q.

The dynamics on the Hamiltonian formalism is given by the following equations:

iξnhωQ − dH ∈ W 0, ξnh ∈ T C.

Moreover, as D is a vector subbundle, it is not hard to prove that

ξnh(x) ∈ Hx , ∀ x ∈ C.

In other words, ξnh(x) ∈ Wx for every x ∈ C .

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Proposition 5.2 Let L be a Lagrangian distribution on the symplectic vector subbun-dle H on C. The following statements are equivalent:

(i) 〈dH(y),Ly〉 = 0, that is, 〈dH(y), X (y)〉 = 0 for every section X of L,(ii) (ξnh)(y) ∈ Ly ,

for every y ∈ C.

Proof Since L ⊂ H we have that 〈dH,L〉 = 0 is equivalent to ωQ(ξnh,L) = 0.Therefore ξnh(y) ∈ L⊥

y = Ly for every y ∈ C because L is a Lagrangian distributionwhere

L⊥ = {V ∈ H |ωQ(V,L) = 0} .

��Moreover, we have that TπQ(H) = D and if γ is a section of the fiber bundle

(πQ)|C : C → Q, then Tγ (D) ∈ H. Thus, if we define the vector field ξγnh on Q by

ξγ

nh(x) = Tγ (x)πQ(ξnh(γ (x))),

then we have the following corollary, as a particular case of Proposition 5.2. (See[16,19])

Corollary 5.3 If Tγ (D) is a Lagrangian distribution on H along the points of γ (Q).The following statements are equivalent:

(i) 〈d(H ◦ γ ), D〉 = 0,(ii) ξnh and ξγnh are γ -related.

Proof The proof follows the same lines as Proposition 5.2. ��Observe that if Tγ (D) is a Lagrangian distribution along γ (Q), then it implies that

for all v1, v2 ∈ D we have that

0 = ωQ(Tγ (v1),Tγ (v2))

= −(γ ∗dθ)(v1, v2)

= −d(γ ∗θ)(v1, v2)

= −dγ (v1, v2)

where we have used the fact that γ ∗θ = γ , see Sect. 2 for more details. This isprecisely the condition that appears in previous approaches to Hamilton–Jacobi theoryfor nonholonomic systems, see [16,19,27,29].

Remark 5.4 If D is completely nonholonomic, that is, the smallest involutive distri-bution under the Lie bracket which contains D is the entire tangent bundle, then thecondition (i) in Proposition 5.2 implies that H ◦ γ is constant. This last condition isanalogous to condition (ii) in Theorem 3.1.

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6 Generalized time-dependent Hamilton–Jacobi theory

From now on, we focus on the geometric version of time-dependent Hamilton–Jacobiequation in terms of Lagrangian submanifolds. We first state a general Hamilton–Jacobi theorem for a time-dependent 1-form and a family of Lagrangian submanifoldsof (T ∗Q, ωQ), extending Sect. 4 to the non-autonomous case.

Let αt be a time-dependent family of 1-forms on T ∗Q and Lt be Lagrangiansubmanifolds of (T ∗Q, ωQ), we define the following family of affine distributions onT T ∗Q fibering over Lt

�t = �(αt |Lt

) + T Lt , (5)

where � : T ∗T ∗Q → T T ∗Q is the musical isomorphism.With these elements we obtain a more general version of time-dependent Hamilton–

Jacobi result than the one stated in Theorem 3.3.

Theorem 6.1 Let Lt be a family of Lagrangian submanifolds of (T ∗Q, ωQ), αt bea family of 1-forms on T ∗Q, (�t )t∈R be defined as in (5) and H be a Hamiltonianfunction on T ∗Q. The following statements are equivalent:

(i) T∗iLt (dH − αt ) = 0 for every t ∈ R, where iLt : Lt → T ∗Q is the canonicalimmersion,

(ii) (dH − αt )|Lt∈ (T Lt )

0,(iii) X H |Lt ∈ �t .

Proof Note that condition (i) is straightforward equivalent to condition (ii). By apply-ing the sharp musical isomorphism to (ii), (dH)|Lt − αt ∈ (T Lt )

0, we have

�((dH − αt )|Lt ) ∈ �((T Lt )0).

Equivalently, we have

�((dH − αt )pq ) ∈ �((Tpq Lt )0) = (Tpq Lt )

⊥ = Tpq Lt ,

for every pq ∈ Lt , because Lt is a Lagrangian submanifold. Thus,

X H |Lt ∈ �(αt |Lt )+ T Lt = �t

and this concludes the proof because all the arguments can be reversed. ��Now let us consider the particular case where the family of Lagrangian submanifolds

Lt is given by a time-dependent family of closed 1-forms on Q. In other words, considera smooth map γ : R × Q → T ∗Q such that γt : Q → T ∗Q is a closed 1-form on Qfor all t ∈ R. Locally, γ (t, q) = γt (q) = (qi , γi (t, q)).

For each t , define the subset Lt = Imγt ⊂ T ∗Q. As in the general case at thebeginning of this section, we cannot only consider the tangent space to Lt as in Sect. 4

123


because of the time-dependence of the family of sections. That is why we define thefollowing family of affine distributions on T T ∗Q fibering over Lt = Imγt :

�t = Tγ

(∂

∂t

)+ Tγt (T Q), (6)

where Tγ : T (R × Q) → T T ∗Q and Tγt : T Q → T T ∗Q. Locally,

�t ={(

qi , γi (t, q), qi ,∂γi

∂t(t, q)+ q j ∂γi

∂q j(t, q)

)∈ T T ∗Q | ∀ (q, q) ∈ T Q

}.

In other words,

(�t )γt (q) = ∂γi

∂t

∂

∂pi γt (q)+ spanR

{∂

∂q j+ ∂γi

∂q j

∂

∂pi

}γt (q)

.

Consider the 1-form αt on T ∗Q defined by

αt =

(Tγ

(∂

∂t

)),

with local expression αt (q, γi (t, q)) = −∂γi

∂t(t, q) dqi . Now we can state the follow-

ing corollary as a particular case of Theorem 6.1.

Corollary 6.2 Let Lt = Imγt ⊂ T ∗Q where γt is a family of closed 1-forms on Qand (�t )t∈R be defined as in (6). Define the family of vector fields Xγt

H ∈ X(Q) by

XγtH = TπQ ◦ X H ◦ γt .

The following statements are equivalent:

(i) d(H ◦ γt )+ ∂γi

∂tdqi = 0;

(ii) d(H)|Lt − αt ∈ (T Lt )0;

(iii) X H |Lt∈ �t ;

(iv) if c : I → Q is an integral curve of XγtH , then γt ◦ c : I → T ∗Q is an integral

curve of X H .

Proof (iii) ⇔ (iv). Locally, (X H )|Lt ∈ �t is equivalent to

− ∂H

∂qi= ∂γi

∂t+ ∂γi

∂q j

∂H

∂p j(7)

along Im γt . To prove (iv) we start with an integral curve c : I → Q of XγtH . If locally

c(t) = (qi (t)), then

dqi

dt= ∂H

∂pi(qi (t), γi (t, q(t))) .

123


Now, using (7) we have that

d (γi ◦ c)

dt= ∂γi

∂t+ ∂γi

∂q j

∂H

∂p j

= −∂H

∂qi.

Therefore, γt ◦ c : I → T ∗Q is an integral curve of X H . The converse can be provedby reversing the arguments.

The remaining equivalences follow straightforward from Theorem 6.1. ��Remark 6.3 Note that for a function W : R × Q → R, we have

R × QdW ��

γ

�� T ∗(R × Q)

π2��

T ∗Q

Then in Corollary 6.2 we have that

γi (t, q)dqi = ∂W

∂qi(t, q)dqi .

Thus condition (i) in Corollary 6.2 can be rewritten as

d

(H ◦ γt − ∂W

∂t

)= 0,

or equivalently

H

(qi ,

∂W

∂qi

)+ ∂W

∂t= K (t)

where K (t) ∈ R, which is the same condition as the one in (ii) in Theorem 3.3.

7 Applications of time-dependent Hamilton–Jacobi theory

Analogously to Sect. 5 we use here particular time-dependent families of theLagrangian submanifolds in (5) to develop the novel applications of Theorem 6.1in non-autonomous Hamilton–Jacobi theory.

7.1 Holonomic time-dependent dynamics

Let N be a submanifold of Q and h be a Hamiltonian function on T ∗N . Take γ : R ×N → T ∗N such that {γt }t∈R is a family of closed 1-forms γt on N . Then, we have that

123


Lt = Im γt is a Lagrangian submanifold of T ∗N and at the same time we can defineanother family of Lagrangian submanifolds LN ,γt of T ∗Q according to (2) which isgiven by

LN ,γt = {μ ∈ T ∗Q | T∗iN (μ) = γt }= {μ ∈ T ∗Q | μiN (x)(Tx iN (vx )) = γt,x (vx ) ∀ vx ∈ Tx N },

where iN : N ↪→ Q is the canonical immersion. This situation is summarized in thefollowing diagram:

R T ∗NπN

��

h T ∗Q

πQ

��

T∗iN LN ,γt� �

iLN ,γt

πQ∣∣LN ,γt

��N

γt

��

N

for every t ∈ R. Hence we can extend the Hamiltonian h to a hamiltonian H :T ∗

N Q −→ R defined by H = h ◦ T∗iN .As described at the beginning of Sect. 6, the time-dependence in the dynamics

forces us to introduce a 1-form αt on T ∗N . In order to describe the holonomic time-dependent Hamilton–Jacobi equation on the manifold T ∗

N Q we must define a familyof 1-forms �t on T ∗

N Q obtained from αt as follows:

〈�t (μq), X (μq)〉 = 〈αt ,Tμq

(T∗iN

)X (μq)〉, (8)

for all μq ∈ T ∗N Q, X (μq) ∈ Tμq T ∗

N Q, Tμq (T∗iN ) : Tμq T ∗

N Q → Tq T ∗N . In fact,�t = T∗(T∗iN ) (αt ) because T∗(T∗iN ) : T ∗(T ∗N ) → T ∗(T ∗

N Q).

Proposition 7.1 Let αt be a time-dependent family of 1-forms on T ∗N, standardtime-dependent Hamilton–Jacobi equation for holonomic dynamics on N

γ ∗t dh − T∗iLt (αt ) = 0, that is, h ◦ γt − αt |Lt

= constant for every t,

is equivalent to

T∗iLN ,γt(dH −�t ) = 0,

where H : T ∗N Q −→ R, H = h ◦ T∗iN and �t is the 1-form on T ∗

N Q defined in (8).

123


Proof For μq ∈ LN ,γt ⊆ T ∗N Q,

0 = 〈T∗iLN ,γt(dH −�t ), Xμq 〉 = 〈(dH −�t )μq ,Tμq iLN ,γt

Xμq 〉= 〈(d (

h ◦ T∗iN) − T∗(T∗iN ) (αt ))μq ,Tμq iLN ,γt

Xμq 〉= 〈d (h ◦ γt )μq

,Tμq iLN ,γtXμq 〉 − 〈T∗(T∗iN ) (αt )μq

,Tμq iLN ,γtXμq 〉

= 〈d (h ◦ γt )μq,Tμq (πQ)

∣∣LN ,γtXμq 〉 − 〈(αt )μq

,TiLN ,γt(μq )(T

∗iN )Tμq iLN ,γtXμq 〉

= 〈d (h ◦ γt )μq,Tμq (πQ)

∣∣LN ,γtXμq 〉 − 〈(αt )μq

,Tq iLt Tμq (πQ)∣∣LN ,γt

Xμq 〉.= 〈(d (h ◦ γt )− T∗iLt (αt )

)μq,Tμq (πQ)

∣∣LN ,γtXμq 〉.

The above equalities are proved using the same arguments as in the proof of Proposi-tion 5.1. From here the result follows. ��

Locally, γt = dWt with W : R× N → R and φα are constraints defining implicitlyN . Consider an arbitrary extension W of W to a neighborhood of R × Q so that

LN ,γt = {μ ∈ T ∗Q | μ = dWt + λt,αdφα}.

Therefore, the Hamilton–Jacobi equation is rewritten as

H(dW (q)+ λt,αdφα(q))+ ∂W

∂t= k(t),

φα(qi ) = 0.

⎫⎬⎭

7.2 Nonholonomic time-dependent mechanics

Having in mind the description of nonholonomic mechanics in Sect. 5 the time-dependent case works as follows. Consider a family of Lagrangian distributions Lt onthe symplectic vector subbundle H on C and a family of 1-forms αt on T ∗Q. Insteadof considering the tangent bundle TLt we must define the following submanifold ofTC T ∗Q:

�t = � (αt |C )+ TLt . (9)

Proposition 7.2 Let Lt be a family of Lagrangian distributions on the symplecticvector subbundle H on C, αt be a family of 1-forms on T ∗Q and (�t )t∈R be definedas in (9). The following statements are equivalent:

(i) 〈(dH − αt ) (y), (Lt )y〉 = 0, that is, 〈dH(y), X (y)〉 = 0 for every section X ofLt ,

(ii) (ξnh)(y) ∈ (�t )y ,

for every y ∈ C and for every time t.

123


Remark 7.3 As in Sect. 5.2, for the time-dependent nonholonomic case it is also nec-essary to adapt the theory developed in Sect. 6 to the framework of symplectic vectorsubbundles by using Lagrangian distributions instead of Lagrangian submanifolds.

Now we consider the particular case of Lagrangian distributions on H defined bya family of sections γt of the fiber bundle (πQ)

∣∣C : C → Q. These sections satisfy

Tγt (D) ∈ H. For this particular case the submanifold �t in (9) becomes

�t = Tγ

(∂

∂t

)+ Tγt (D).

If we also define the vector field ξγtnh on Q by

ξγtnh(x) = Tγt (x)πQ(ξnh(γt (x))),

then we have the following corollary as a particular case of Proposition 7.2.

Corollary 7.4 If Tγt (D) is a Lagrangian distribution on H along the points of γt (Q).The following statements are equivalent:

(i)

⟨d(H ◦ γt )+ ∂γi

∂tdqi , D

⟩= 0,

(ii) ξnh |Tγt (D) ⊆ �t ,(iii) ξnh and ξγt

nh are γt -related.

Proof The proof follows the same lines as the ones of Proposition 5.2 and Corol-lary 6.2. ��

8 Conclusions and future developments

In this paper we have described a general version of Hamilton–Jacobi equation whichhas allowed us to consider different examples of mechanical systems such as holo-nomic, nonholonomic or time-dependent systems within the same framework. Ourapproach has been based on the notion of Lagrangian submanifolds as a tool to con-cisely prove a version of the Hamilton–Jacobi theorem and to give special emphasison applications. In particular, we have derived a Hamilton–Jacobi equation for holo-nomic systems in such a way that the associated Lagrange multipliers has been pro-vided with a geometrical interpretation. Moreover, using the “symplectic approach”to nonholonomic mechanics (see [5,25]) we have shown that our techniques are alsoadapted to the nonholonomic version of Hamilton–Jacobi equation. This result is moregeneral than those known in the literature. Finally, using time-dependent families ofLagrangian submanifolds we have directly obtained the corresponding versions of thetime-dependent Hamilton–Jacobi equation.

One of the main advantages of this paper is the possibility to use the same versionof the Hamilton–Jacobi equation to deal with very different situations such as time-dependency, presence of constraints, etc. Moreover, it is possible to adapt this frame-work to other cases like Hamilton–Jacobi equation for singular Lagrangian systems,

123


reduced systems, systems whose dynamics are described by Poisson or almost-Poissonbrackets and, even for classical field theories. It is also envisaged that the approachin this paper would contribute to getting new insights into the study of caustics ofLagrangian submanifolds [3,4].

Acknowledgments This work has been partially supported by MICINN (Spain) MTM2008-00689,MTM2010-21186-C02-01 and MTM2010-21186-C02-02; 2009SGR1338 from the Catalan government;the European project IRSES-project GeoMech-246981 and the ICMAT Severo Ochoa project SEV-2011-0087. MBL has been financially supported by Juan de la Cierva fellowship from MICINN. The authors araalso grateful to the referees for their useful comments.

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