28049 Madrid, Spain. (Communicated by Andrew Lewis). Abstract. In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds.
JOURNAL OF GEOMETRIC MECHANICS c
American Institute of Mathematical Sciences Volume 6, Number 1, March 2014
doi:10.3934/jgm.2014.6.121 pp. 121–140
A HAMILTON-JACOBI THEORY ON POISSON MANIFOLDS
´ n, David Mart´ın de Diego and Miguel Vaquero Manuel de Leo Instituto de Ciencias Matem´ aticas (CSIC-UAM-UC3M-UCM) c\ Nicol´ as Cabrera, n 13-15, Campus Cantoblanco, UAM 28049 Madrid, Spain
(Communicated by Andrew Lewis) Abstract. In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
1. Introduction. The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q i ) (called the principal function) such that ∂S ∂S + h(q i , i ) = 0, (1) ∂t ∂q where h = h(q i , pi ) is the hamiltonian function of the system. If we put S(t, q i ) = W (q i ) − tE, where E is a constant, then W satisfies ∂W h(q i , i ) = E; (2) ∂q W is called the characteristic function. Equations (1) and (2) are indistinctly referred as the Hamilton-Jacobi equation (see [1, 2, 22]). The Hamilton-Jacobi equation helps to solve the Hamilton equations for h ∂h dpi ∂h dq i = = − i. (3) , dt ∂pi dt ∂q Indeed, if we find a solution W of the Hamilton-Jacobi equation (2) then any solution (q i (t)) of the first set of equations (3) gives a solution of the Hamilton equations by taking pi (t) = ∂W ∂q i . This result can be founded in [1]. Moreover, one can rephrase the above result by stating that if W is a solution of the Hamilton-Jacobi equation, then dW (a 1-form on Q) transforms the integral curves of the vector field XhdW = T πQ ◦ Xh ◦ dW into the integral curves of Xh ; here, Xh is the Hamiltonian vector field defined by the hamiltonian h and πQ : T ∗ Q −→ Q is the canonical projection. Of course we can 2010 Mathematics Subject Classification. Primary: 70H20, 70 G45, 70F25, 37J60. Key words and phrases. Hamilton-Jacobi theory, Poisson manifolds, nonholonomic mechanics, time-dependent systems, external forces. This work has been partially supported by MICINN (Spain) MTM 2010-21186-C02-01, MTM 2009-08166-E, the European project IRSES-project “Geomech-246981” and the ICMAT Severo Ochoa project SEV-2011-0087. M. Vaquero wishes to thank MINECO for a FPI-PhD Position. The authors are also grateful to the referees for their useful comments.
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think in a more general situation where we look for general 1-forms on Q that play a similar role to dW . This geometrical procedure has been succesfully applied to many other different contexts, including nonholonomic mechanics (see [7, 8, 11, 13]), singular lagrangian systems [15, 16], and even classical field theories [12, 17, 14]. Notice that in these frameworks, we don’t have a symplectic framework; for instance, in nonholonomic mechanics the natural geometric framework is provided by a (2,0)-tensor field (an almost Poisson tensor) on the constraint submanifold that it is not integrable (that is, it is not satisfies the Jacobi identity). The almost-Poisson bracket in nonholonomic mechanics has been firstly introduced by A. van der Shaft and B.M. Mashke ([24]). All these scenarios are just the motivation for the investigation developed in this paper. Our goal is to develop a Hamilton-Jacobi theory in a unifying and more general setting, say hamiltonian systems on an almost-Poisson manifold, that is, a manifold equipped with a skew-symmetric (2, 0)-tensor field which does not necesarily satisfies the Jacobi identity. We also assume that the almost-Poisson manifold has a fibered structure over another manifold. The Hamilton-Jacobi problem now is to find a section of the fibered manifold such that its image is a lagrangian submanifold and the differential of the given hamiltonian vanishes on the tangent vectors to the section and belonging to the characteristic distribution. The theory includes the case of classical hamiltonian systems on the cotangent bundle of the configuration manifold as well as the case of nonholonomic mechanical systems. We also apply the theory to time-dependent hamiltonian systems and systems with external forces. We also discuss the existence of complete solutions and prove that if a complete solution exists then we obtain first integral in involution, which is a remarkable fact since our framework is just almost-Poisson. Along the paper, all the manifolds are real, second countable and C ∞ . The maps are assumed to be also C ∞ . Sum over all repeated indices is understood 2. Hamilton-Jacobi theory on almost-Poisson manifolds. 2.1. Hamiltonian systems on almost-Poisson manifolds. Assume that (E, Λ) is an almost-Poisson manifold, that is, E is a manifold equipped with an almostPoisson structure Λ, which means that Λ is a skew-symmetric (2, 0)-tensor field on E. Notice that Λ does not necessarily satisfy the Jacobi identy; in that case, we will have a Poisson tensor, and E will be a Poisson manifold. For the moment, one only needs to ask (E, Λ) be an almost-Poisson manifold. Therefore, Λ defines a vector bundle morphism ] : T ∗ E −→ T E by h](α), βi = Λ(α, β) ∗
for all α, β ∈ T E. Of course, we shall also denote by ] the induced morphism of C ∞ -modules between the spaces of 1-forms and vector fields on E. Notice that we will use the notation ]Λ if there is danger of confusion. We denote by C the characteristic distribution defined by Λ, that is Cp = ](Tp∗ E)
A HAMILTON-JACOBI THEORY ON POISSON MANIFOLDS
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for all p ∈ E (in other terms, Cp = Im]p , where ]p = Tp∗ E −→ Tp E). The rank of the almost-Poisson structure at p is the dimension of the space Cp . Notice that C is a generalized distribution and, moreover, is not (in general) integrable since Λ is not Poisson in principle. The following lemma will be useful Lemma 2.1. Let (E, Λ) be an almost-Poisson manifold, then we have C ◦ = ker(]), where C ◦ denotes the annihilator of C. Proof. Observe that ◦
= {µ ∈ Tp∗ E | hµ, ]p (α)i = 0, ∀α ∈ Tp∗ E}
(Im]p )
= {µ ∈ Tp∗ E | h]p (µ), αi = 0, ∀α ∈ Tp∗ E} =
ker ]p
and thus, the result holds. We also have the following definition Definition 2.2. ([19, 23]) A submanifold N of E is said to be a lagrangian submanifold if the following equality holds ](T N ◦ ) = T N ∩ C. To have dynamics we need to introduce a hamiltonian function h : E −→ R, and thus we obtain the corresponding hamiltonian vector field Xh = ](dh). 2.2. Hamilton-Jacobi theory on almost-Poisson manifolds. Assume now that the almost-Poisson manifold E with almost-Poisson tensor Λ fibres over a manifold M , say π : E −→ M is a surjective submersion (in other words, a fibration). Assume that γ is a section of π : E −→ M , i.e. π ◦ γ = idM . Define the vector field Xhγ on M by Xhγ = T π ◦ Xh ◦ γ. The following diagram summarizes the above construction:
