Formulation of Stochastic Contact Hamiltonian Systems
aa r X i v : . [ m a t h . D S ] F e b Formulation of Stochastic Contact Hamiltonian Systems
Pingyuan Wei a) and Zibo Wang b) School of Mathematics and Statistics & Center for Mathematical Sciences,Huazhong University of Science and Technology, Wuhan 430074,China (Dated: 18 February 2021)
In this work we devise a stochastic version of contact Hamiltonian systems, andshow that the phase flows of these systems preserve contact structures. Moreover,we provide a sufficient condition under which these stochastic contact Hamiltoniansystems are completely integrable. This establishes an appropriate framework forinvestigating stochastic contact Hamiltonian systems.Keywords: Contact Hamiltonian systems, contact vs. symplectic structures, preserv-ing contact structure, completely integrable.
It is well known that contact Hamiltonian systems are the “odd-dimensional cousins” of symplectic Hamiltonian systems. Contact Hamil-tonian systems arise in the description of dissipative systems, and havefound applications in many areas of science. Motivated by Bismut’s workabout stochastic Hamiltonian systems on symplectic manifolds that pre-serve symplectic structures, we propose a stochastic version of contactHamiltonian systems which preserve contact structures. We also analysewhen a stochastic contact Hamiltonian system is completely integrable.Two examples are presented to illustrate our results. a) Electronic mail: [email protected] b) Electronic mail: (Corresponding author) [email protected] . INTRODUCTION Hamiltonian systems on symplectic manifolds are the natural framework for classical me-chanics and statistical physics, and the starting point for their quantum counterparts . Asmodels for conservative systems, they have been extensively studied . Hamiltonian systemson contact manifolds are an appropriate scenario for certain dissipative systems . ContactHamiltonian systems are not only an odd dimensional counterpart of symplectic cases, butalso contain rich geometric structures. In recent years, contact Hamiltonian systems haveattracted a lot of attentions. For example, they are used for the study of thermodynamics ,optimal control and fluid mechanics . Some theories such as weak KAM theory , completeintegrability and variational principle for contact Hamiltonian systems have been de-veloped recently. Contact algorithms are also proposed to simulate such systems .Stochastic dynamical systems can be regarded as models that take into account the influ-ence of the random environment . There are recent studies on Hamiltonian systems withrandom perturbations; for example, Brin-Freidlin and MacKay . A stochastic Hamil-tonian system preserving symplectic structure was proposed by Bismut . Following thisframework, Li and Wei discussed averaging principles for completely integrable stochasticHamiltonian systems. Moreover, the symplectic structure-preserving numerical algorithmsare developed. For example, Wang et al. designed a stochastic variational integrator usingHamilton’s principle.Integrability is a topic worth discussing in dynamical systems. The famous Arnold-Liouville theorem describes a connection between completely integrable Hamiltonian systemsand toric geometry in the symplectic setting . Under the hypothesis of Arnold-Liouville the-orem, a Hamiltonian system can be expressed by the action-angle coordinate, such that afirst integral only depends on the action variable, while the angle variable changes on torus .As mentioned earlier, a contact version of complete integrability has been stuided recently.In this paper, we devise a class of stochastic contact Hamiltonian systems which preservecontact structures. Then we provide a sufficient condition under which these stochasticcontact Hamiltonian systems are completely integrable.This paper is organized as follows. In section 2, we present a class of stochastic contactHamiltonian systems. In section 3, we demonstrate that the phase flow of these systems pre-serve contact structures. The key idea of the proof is a chain rule that holds for Stratonovitch2ifferentials. In section 4, we show that a class of stochastic contact Hamiltonian systemsare completely integrable. Finally, we present two illustrative examples. In particular, weestablish the generalized contact action-angle coordinates in the second example.
2. STOCHASTIC CONTACT HAMILTONIAN SYSTEMS
A smooth (2 n + 1)-dimensional differential manifold M is said to be a contact manifold ifit is equipped with a contact structure, which is a nondegenerate 1-form η such that η ∧ ( dη ) n is a volume form (i.e., it is nonzero at each point of M ).Given a smooth (Hamiltonian) function H and a family of d smooth (Hamiltonian)functions { H k } dk =1 on M . The corresponding contact Hamiltonian vector fields, denoted by X H and X H k ( k = 1 , , ..., d ), are defined through the following intrinsic relations H i = ι X Hi η and dH i = − ι X Hi dη + R ( H i ) η (2.1)for i = 0 , , · · · , d , where R is Reeb vector field (which is the unique vector field such that ι R η = 1 and ι R dη = 0).We propose the following stochastic contact Hamiltonian system: dx t = X H ( x t ) dt + d X k =1 X H k ( x t ) ◦ dB kt , x ( t ) = x ∈ M, (2.2)where { B kt } dk =1 are pairwise independent Brownian motions on a probability space (Ω , F , P )and “ ◦ ” stands for Stratonovitch differentials. We call X H the drift vector field and X H k the diffusion vector fields. Remark . Notice that we have chosen to write (2.2) in Stratonovich rather than Itˆostochastic differentials. This is mainly because Stratonovich differentials has the advantageof leading to ordinary chain rule of the Newton-Leibniz type under coordinates transforma-tion. Such a property offers some reduction in calculations, and makes the Stratonovichdifferential natural to use especially in connection with stochastic differential equations onmanifolds. We also note that Brownian motions in (2.2) can be extended to more gen-eral stochastic processes, for example L´evy processes or semimartingales. Accordingly, theStratonovich differentials should be replaced by Marcus differentials.
Remark . Formally, equation (2.2) can be rewritten in the form of a Hamiltonian system ddt x t = X ˜ H ( x t ) with a “randomized” contact Hamiltonian ˜ H = H + P dk =1 H k w kt , where w t
3s an d -dimensional noise (which is a Gaussian white noise for the Brownian case). One mayregard this as a contact Hamiltonian within the external world. A “deterministic” contactHamiltonian system with Hamiltonian H is considered as a dissipative mechanical system.The stochastic part in (2.2) is introduced to characterize the complicated interaction betweenthe deterministic system and the fluctuating environment. Nevertheless, in the next section,we will show that the stochastic flow for equation (2.2) still preserves the contact strucure.If the stochatsic part in (2.2) takes other forms, the corrsponding stochastic flow may notpreserve the contcat structure.In view of some concrete applications, it is convenient to write system (2.2) in local coordi-nates. According to Darboux theorem for contact manifolds, around each point in M , onecan find local (called Darboux or canonical) coordinates ( q, p, z ) = ( q , · · · , q n , p , · · · , p n , z )such that η = dz − pdq, R = ∂∂z . (2.3)Here pdq has to be read as P nj =1 p j dq j . This same shorthand notation applies to analogousexpressions below.In Darboux coordinates, we get the following local expressions for Hamilton vector fields X H i = ∂H i ∂p ∂∂q − (cid:18) ∂H i ∂q + p ∂H i ∂z (cid:19) ∂∂p + (cid:18) p ∂H i ∂p − H i (cid:19) ∂∂z , (2.4)for i = 0 , , · · · , d . Therefore, a canonical stochastic contact Hamiltonian system can bewritten as dq = ∂H ∂p dt + d X k =1 ∂H k ∂p ◦ dB kt , (2.5) dp = − (cid:18) ∂H ∂q + p ∂H ∂z (cid:19) dt − d X k =1 (cid:18) ∂H k ∂q + p ∂H k ∂z (cid:19) ◦ dB kt , (2.6) dz = (cid:18) p ∂H ∂p − H (cid:19) dt + d X k =1 (cid:18) p ∂H k ∂p − H k (cid:19) ◦ dB kt (2.7)with initial state ( q ( t ) , p ( t ) , z ( t )) , ( α, β, γ ), t >
3. CONFORMAL CONTACTOMORPHISM
As in the deterministic case , we are interested in the system (2.5)-(2.7) such that thesolution mapping ( α, β, γ ) → ( p, q, z ) preserves the natural contact structure (leaves the4ontact form invariant) up to multiplication by a conformal factor. That is, the phase flow φ t of the system is a conformal contactomorphism in the sense φ ∗ t η = λ t η (3.1)for a nowhere zero function (i.e., conformal factor) λ t : M → R . In local coordinates, weonly need to focus on dz − pdq = λ t ( dγ − βdα ) . (3.2)Note that the conformal factor λ t , with t ∈ [ t , ∞ ), is a smooth family with λ t = 1. Henceall λ t take value in R + . To avoid confusion, we should note that the differentials in (2.5)-(2.7) and (3.2) have different meanings: In (2.5)-(2.7), p, q, z are treated as functions of time,while, in (3.2), the differentiation is made with respect to the initial data α, β, γ . Remark . For deterministic case, i.e., ˙ x = X H ( x ), the corresponding phase flow ψ t isa conformal contactomorphism. Indeed, by Cartan’s identity and (2.1), we have L X H η = −R ( H ) η . Note that ∂∂t ψ ∗ t η = ψ ∗ t L X H η = ψ ∗ t ( −R ( H ) η ) = ( −R ( H ) ◦ ψ t ) ψ ∗ t η . We concludethat ψ ∗ t η = λ t η with λ t = exp (cid:0) − R tt ( R ( H ) ◦ ψ τ ) (cid:1) dτ . We also remark that a map f is calleda (strict) contactomorphism if f ∗ η ≡ η . Theorem 3.2. (Preserving the Contact Structure)
The phase flow of the stochasticcontact Hamiltonian system (2.5) - (2.7) is a conformal contactomorphism (i.e., it preservesthe contact structure) with a conformal factor λ t = exp (cid:0) − Z tt ∂∂z H dτ − d X k =1 Z tt ∂∂z H k ◦ dB kτ (cid:1) . (3.3) Proof.
Notice that the condition (3.2) for phase flow to be conformal contactomorphism canbe rewritten as ∂z∂α r − P nj =1 p j ∂q j ∂α r = − λ t β r , r = 1 , · · · , n, ∂z∂β r − P nj =1 p j ∂q j ∂β r = 0 , r = 1 , · · · , n, ∂z∂γ − P nj =1 p j ∂q j ∂γ = λ t . (3.4)For convenience, we adopt the following notation (for j, r = 1 , · · · , n ): q jrα = ∂q j ∂α r , q jrβ = ∂q j ∂β r , q jγ = ∂q j ∂γ . (3.5)Similarly, we define p jrα , p jrβ , p jγ , z rα , z rβ and z γ .5y calculating at ( q, p, z ) = ( q ( t ; t , α, β, γ ) , p ( t ; t , α, β, γ ) , z ( t ; t , α, β, γ )) which is a so-lution to system (2.5)-(2.7), we conclude that p jγ , q jγ , z γ ( j = 1 , · · · , n ) satisfy the followingsystem of stochastic differential equations dq jγ = n X l =1 ( ∂ H ∂p j ∂q l q lγ + ∂ H ∂p j ∂p l p lγ + ∂ H ∂p j ∂z z γ ) dt + d X k =1 n X l =1 ( ∂ H k ∂p j ∂q l q lγ + ∂ H k ∂p j ∂p l p lγ + ∂ H k ∂p j ∂z z γ ) ◦ dB kt , (3.6) dp jγ = − n X l =1 h(cid:16) ∂ H ∂q j ∂q l + p j ∂ H ∂z∂q l (cid:17) q lγ + (cid:16) ∂ H ∂q j ∂p l + δ jl ∂H ∂z + p j ∂ H ∂z∂p l (cid:17) p lγ + (cid:16) ∂ H ∂q j ∂z + p j ∂ H ∂z (cid:17) z γ i dt − d X k =1 n X l =1 h(cid:16) ∂ H k ∂q j ∂q l + p j ∂ H k ∂z∂q l (cid:17) q lγ + (cid:16) ∂ H k ∂q j ∂p l + δ jl ∂H k ∂z + p j ∂ H k ∂z∂p l (cid:17) p lγ + (cid:16) ∂ H k ∂q j ∂z + p j ∂ H k ∂z (cid:17) z γ i ◦ dB kt , (3.7) dz γ = n X l =1 h(cid:16) n X j =1 p j ∂ H ∂p j ∂q l − ∂H ∂q l (cid:17) q lγ + (cid:16) n X j =1 p j ∂ H ∂p j ∂p l (cid:17) p lγ + (cid:16) n X j =1 p j ∂ H ∂p j ∂z − ∂H ∂z (cid:17) z γ i dt + d X k =1 n X l =1 h(cid:16) n X j =1 p j ∂ H k ∂p j ∂q l − ∂H k ∂q l (cid:17) q lγ + (cid:16) n X j =1 p j ∂ H k ∂p j ∂p l (cid:17) p lγ + (cid:16) n X j =1 p j ∂ H k ∂p j ∂z − ∂H k ∂z (cid:17) z γ i ◦ dB kt , (3.8)with initial state (( q jrα ) , ( p jrα ) , z rα ) = ( δ jr , , q ( t ) , p ( t ) , z ( t )) = ( α, β, γ ), it is clear that λ t = 1.The third condition in (3.4) is fulfilled if and only if dz γ ( t ) − X j dp j ( t ) · q jγ ( t ) − X j p j · dq γ ( t ) = dλ t . (3.9)Due to (2.6), (3.6) and (3.8), we conclude that the relation (3.9) becomes (cid:16) z γ − X j p j q rγ (cid:17) ∂H ∂z dt + d X k =1 ∂H k ∂z ◦ dB kt ! = − dλ t . (3.10)6imilarly, the first and the second conditions in (3.4) are fulfilled if and only if (cid:16) z rα − X j p j q jrα (cid:17) ∂H ∂z dt + d X k =1 ∂H k ∂z ◦ dB kt ! = β r dλ t , (3.11) (cid:16) z rβ − X j p j q jrβ (cid:17) ∂H ∂z dt + d X k =1 ∂H k ∂z ◦ dB kt ! = 0 . (3.12)As a consequence, the condition (3.4) holds if and only if λ t ∂H ∂z + d X k =1 ∂H k ∂z ◦ ˙ B kt ! = − ˙ λ t , (3.13)with λ t = 1. By solving equation (3.13), we have λ t = exp (cid:0) − Z t ∂∂z H ( q τ , p τ , z τ ) dτ − d X k =1 Z t ∂∂z H k ( q τ , p τ , z τ ) ◦ dB kτ (cid:1) . This implies that (3.4) holds and hence conformal contactomorphism condition (3.2) isfulfilled. The proof is complete.
4. COMPLETE INTEGRABILITY
In Bismut’s setting, a class of stochastic completely integrable (symplectic) Hamiltoniansystems has been studied. See more information in Li . The integrability for contactHamiltonian systems has also drawn much recent attention .Note that although any smooth function can be chosen as a contact Hamiltonian, it isoften convenient to choose the function 1 = ι R η as the Hamiltonian, making the Reeb vectorfield R the Hamiltonian vector field. The Hamiltonian contact structure in this case is saidto be of Reeb type . From now on, we will focus on this case and introduce the correspondingcomplete integrability conditions.Similar to the Poisson bracket in symplectic geometry, the Lie algebra structure of C ∞ ( M )in contact Hamiltonian setting is given by the Jacobi bracket[ f, g ] η = − ι X f ι X g dη + f ι R dg − gι R df. (4.1)It is worth mentioning that this bracket is bilinear, antisymmetric, and satisfies the Jacobiidentity. Furthermore, it fulfils a “weak” Leibniz rule:[ f, gh ] η = [ f, g ] η h + g [ f, h ] η − [ f, η , (4.2)7hich is the key difference with the Poisson bracket.If a function h satisfies that [ h, η = − ι R dh = 0, it represents a first integral of the vectorfield R . A contact Hamiltonian structure of Reeb type is said to be completely integrable ifthere exists ( n + 1) first integrals h = 1 , h , · · · , h n that are independent and in involution.That is, the corresponding Hamiltonian vector fields are linearly independent at almost allpoints and [ h i , η = [ h i , h j ] η = 0 , i, j = 1 , , · · · , n. (4.3)We remark that definition of complete integrability can be generalized to any Hamiltonian.See Boyer for details. We point out that, for the contact form η and a given Hamiltonian H , a function f commute with H may not be equivalent to f being a first integral, unless H is constant along the flow of the Reeb vector field.We also note that a completely integrable contact Hamiltonian system is said to be oftoric type, if the corresponding vector fields X h = R , X h , · · · , X h n form the Lie algebraof a torus T n +1 . Analogous to that of the celebrated Arnold-Liouville theorem in the sym-plectic setting, it is possible to introduce the generalized action-angle variables in contactmanifolds. The action of a torus T n +1 on a (2 n + 1)-dimensional contact manifold ( M, η ) iscompletely integrable if it is effective and preserve the contact structure .We call (2.2) a stochastic completely integrable contact Hamiltonian system, if the familyof d (for convenience, we set d = n + 1 here) smooth contact Hamiltonians { H k } dk =1 form acompletely integrable system in the sense that they are independent and in involution, andif H commutes with this family under Jacobi bracket. Example 4.1. (A stochastic dissipative mechanical system)
We consider the productmanifold M = R × T ⋆ R endowed with the canonical contact structure η = z − p dq − p dq ,where z and ( q, p ) = ( q , q , p , p ) are global coordinates on R and T ⋆ R , respectively. Let H : M → R be a contact Hamiltonian function given by H ( q, p, z ) = 12 m ( p + p ) + V ( q ) + γz, (4.4)with V ( q ) a potential function and γ a positive constant. This Hamiltonian correspondsto a system with a friction force that depends linearly on the velocity (in this case, on the8omentum). When such a contact Hamiltonian system is perturbed by noise, we have thefollowing equation: dq i dt = p i m , dp i dt = − ∂V∂q i − γp i , i = 1 , , and dzdt = 1 m ( p + p ) − H + ε ˙ B t (4.5)or, equivalently, m ¨ q i + γm ˙ q i + ∂V∂q i = 0 , i = 1 , , and dzdt = 12 m ( ˙ q + ˙ q ) − V ( q ) − γz + ε ˙ B t , (4.6)where ˙ B t is a Gaussian white noise and ε is the noise intensity. With H = H and H = ε ,equation (4.5) or (4.6) is indeed a stochastic contact Hamiltonian system. The stochasticflow preserve the contact structure with conformal factor λ t = e − γ ( t − t ) . Furthermore, thefunctions H and H are in involution, i.e., [ H , H ] η = 0. Example 4.2. (A stochastic contact Hamiltonian system in Sasaki-Einstein space)
The homogeneous toric Sasaki-Einstein Space T , was considered as the first example oftoric Sasaki-Einstein/quiver duality and provides supersymmetric backgrounds relevant forthe Ads/CFT correspondence . The space T , can be regarded as a U (1) bundle over S × S . We denote by ( θ i , φ i ), i = 1 , , the coordinates which parametrize the two spheres S in the conventional way, while the angle ψ ∈ [0 , π ) parametrizes the U (1) fiber.The globally defined contact 1-form η SE can be written in the form η SE = 13 ( dψ + cosθ dϕ + cosθ dϕ ) . (4.7)The Reeb vector field defined by η SE is R SE = 3 ∂∂ψ . (4.8)The contact Hamiltonian vector field X H k with Hamiltonian H k can be expressed as X H k =3 X i θ i (cid:18) ∂H k ∂ϕ i − ∂H k ∂ψ cos θ i (cid:19) ∂∂θ i − X i θ i ∂H k ∂θ i ∂∂ϕ i + 3 H k + X i cos θ i sin θ i ∂H k ∂θ i ! ∂∂ψ . (4.9)Consider the following stochastic contact Hamiltonian system d θ θ ϕ ϕ ψ = dt + θ
00 0 0 0 θ ϕ ϕ ◦ dB t , (4.10)9here B t is a 5-dimensional Brownian motion. We find that the corresponding Hamiltonians H = H = 1, H = cos θ , H = cos θ , H = ϕ and H = ϕ , are independent andin involution. In particular, the constant Hamiltonian 1 corresponds to the Reeb vectorfield R SE in (4.8). Therefore, equation (4.10) is a stochastic completely integrable contactHamiltonian system and the stochastic flow preserves the contact structure.Moreover, using the analysis from Visinescu , we can choose T as a compact connectedcomponent of the level set { H = 1 , H = c , H = c } , where c , c are positive constants.Then T is diffeomorphic to a T torus. Let D be an open domain in R . There exist aneighborhood U of T and a diffeomorphism Φ : U → T × D Φ( x ) = ( ϑ , ϑ , ϑ , y , y ) . (4.11)More precisely, we can introduce the angle variables ϑ = ψ , ϑ = ϕ , ϑ = ϕ , (4.12)and the generalized action variables y = 1 , y = 13 cos θ , y = 13 cos θ . (4.13)Therefore, equation (4.10) becomes simplier in these coordinates: d y y ϑ ϑ ϑ = dt + ϑ ϑ ◦ dB t . (4.14)Note that the contact form has the canonical expression η = (Φ − ) ∗ η SE = P i =0 y i dϑ i . Wecall the local coordinates ( ϑ i , y i ) the generalized contact action-angle coordinates. DATA AVAILABILITY
The data that support the findings of this study are available within the article.10
CKNOWLEDGMENTS
The authors would like to thank Prof Maosong Xiang, Dr Lingyu Feng, Dr Jianyu Hu, DrLi Lv and Dr Jun Zhang for helpful discussions. This work was partly supported by NSFCgrants 11771449 and 11531006.
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