The flow method for the Baker-Campbell-Hausdorff formula exact results

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The flow method for the

Baker-Campbell-Hausdorff formula:

exact results

Federico Zadra ∗1, Alessandro Bravetti †2, Angel Alejandro

García-Chung ‡3,4, and Marcello Seri §1

1Bernoulli Institute for Mathematics, Computer Science and Artiﬁcial Intelligence, University

of Groningen, Groningen, The Netherlands

2Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS–UNAM),

Mexico City, Mexico

3Tecnológico de Monterrey, Escuela de Ingeniería y Ciencias, Carr. al Lago de Guadalupe Km.

3.5, Estado de Mexico 52926, Mexico.

4Max Planck Institute for Mathematics in the Sciences Inselstraße 22, 04103 Leipzig, Germany

Leveraging techniques from the literature on geometric numerical integration, we

propose a new general method to compute exact expressions for the BCH formula.

In its utmost generality, the method consists in embedding the Lie algebra of interest

into a subalgebra of the algebra of vector ﬁelds on some manifold by means of an

isomorphism, so that the BCH formula for two elements of the original algebra can

be recovered from the composition of the ﬂows of the corresponding vector ﬁelds.

For this reason we call our method the ﬂow method. Clearly, this method has great

advantage in cases where the ﬂows can be computed analytically. We illustrate its

usefulness on some benchmark examples where it can be applied directly, and discuss

some possible extensions for cases where an exact expression cannot be obtained.

Contents

1 Contact geometry 3

1.1 Contact hamiltonian ﬂows and Jacobi structures .............. 4

2 The ﬂow method 6

∗f.zadra@rug.nl 0000-0001-7565-3792

†alessandro.bravetti@iimas.unam.mx 0000-0001-5348-9215

‡alechung@tec.mx 0000-0002-8982-3569

§m.seri@rug.nl 0000-0002-8563-5894

arXiv:2210.11155v2 [math-ph] 28 Aug 2023

Page 2

3 Closed-form expressions for the BCH formula for various algebras 11

3.1 The Heisenberg algebra ............................ 11

3.2 The contact Heisenberg algebra ....................... 12

3.3 The quadratic symplectic algebra ...................... 13

3.4 The quadratic contact algebra ........................ 15

3.5 The complexiﬁed su(2) ............................ 17

3.6 Splitting numerical integrators ........................ 18

4 Conclusions 20

A Matrix representations 22

A.1 Matrix representation of the CHA ...................... 22

A.2 Matrix representation of QCA ........................ 23

Introduction

The Baker-Campbell-Hausdorﬀ (BCH) formula is a well known result in the theory of Lie

groups and Lie algebras [1,19,26] that found wide application in both mathematics and

physics: from Lie theory [26] to numerical integration [14,37], quantum mechanics [5,

36] and statistical mechanics [39]. Roughly speaking, it links the composition between

two elements of a Lie group in a connected neighbourhood of the identity to an element

of the Lie algebra whose exponential map is the group element which results from the

composition. In other words, it provides a means to compute the map Z:g×g→g

deﬁned by

Z(A, B) := log(eAeB),(1)

where eA, eB∈Gare the corresponding elements in the associated Lie group given by

the exponential map. When the Lie algebra is commutative, the BCH formula is simply

Z(A, B) = A+B.

However, Lie algebras are in general not commutative and ﬁnding closed-form expressions

or even accurate estimates for Zcan become a daunting task. The general BCH formula

can be expressed (formally) in terms of the integral formulation

Z(A, B) = A+B−Z1

∞

n=1

(I−eadAet adB)n

n(n+ 1) dt , (2)

where adAis the adjoint representation of Aand Iis the identity matrix. To approxi-

mate (2), the most common method is to employ the series expansion

Z(A, B) = A+B+ [A, B] + 1

12 ([A, [A, B]] + [B, [B, A]]) + ···

For a detailed treatment of the topic, we refer the reader to [19,26,42].

The problem of computing a closed form for (2) for a given particular algebra has

already been addressed in various settings, employing a number of diﬀerent algebraic

and analytic techniques [6,12,19,31,32,40,45]. In this paper, we present a dynamic

approach to such problem: we compute closed forms of the BCH formula by exploiting an

isomorphism between the given Lie algebra and an appropriate subalgebra of the inﬁnites-

imal contactomorphisms on a contact manifold. The latter in turn is also isomorphic to

Page 3

the Lie algebra of the Hamiltonian functions endowed with the Jacobi bracket that nat-

urally arises from the contact structure [29]. This helps us to simplify the calculations.

The mapping between functions and vector ﬁelds is always an isomorphism for contact

Hamiltonians [29], diﬀerently from the symplectic case [18]. Therefore, even in the case

of some algebras which arise naturally in the symplectic setting (e.g. the Heisenberg

group), the use of the contact structure is more natural, as it guarantees a one-to-one

correspondence between contact Hamiltonian functions and contact Hamiltonian vector

ﬁelds.

We prove the usefulness of our method using some interesting examples arising from

the recent literature in contact geometry and Hamiltonian systems, namely, the Heisen-

berg algebra [26], the contact Heisenberg algebra [12], the quadratic symplectic and the

quadratic contact algebras, and the complexiﬁcation of su(2). Furthermore, we also

include an analysis of contact splitting numerical integrators [14,48] for the damped

harmonic oscillator [11]. More precisely, in this example our method will be used in order

to obtain a priori estimates of the numerical errors for diﬀerent splitting integrators.

The paper is organized as follows: in Section 1we brieﬂy review some concepts in

contact geometry and contact Hamiltonian mechanics. In Section 2we describe the ﬂow

method. Section 3is devoted to the aforementioned examples, presented in detail, while

Section 4presents the conclusions and some additional directions for further research.

Additional results, including a matrix representation of the algebras treated in the paper,

are presented at the end of the paper in Appendix A.

1 Contact geometry

We summarize here the essential concepts of contact geometry that are needed for this

work and refer the reader to [4,7,10,11,13,23,24,29,33] for additional details,

including the relevant proofs. The main deﬁning object for us is a special diﬀerential

1-form, called the contact form, which induces both a volume form on the manifold and a

maximally non-integrable distribution on its tangent bundle, called the contact structure.

Definition 1.1.A (exact) contact manifold is a couple (M, η), where Mis 2n+ 1-

dimensional manifold and ηis a 1-form such that

η∧(dη)n̸= 0.

In this case ηis called a contact form and its kernel at each point deﬁnes a maximally

non-integrable distribution of hyperplanes D, called the contact structure.

If, on the same manifold M, the contact form ηis replaced by η′=fη, for some

nowhere-vanishing real function f, the new 1-form is again a contact form and deﬁnes

the same contact structure. Indeed, the induced volume form is simply

η′∧(dη′)n=fn+1η∧(dη)n̸= 0.

At each p∈M, the contact 1-form ηpand its diﬀerential dηpsplit the tangent space

TpMinto the so-called vertical and the horizontal distributions:

p= Horp(η) = ker(ηp),Vertp(η) = ker(dηp),

where ker(dηp)is the subspace of TpMof all those vectors that annihilate dηp. Thus

the tangent bundle T M can be written as a Whitney sum T M =D

pLVertp(η). It is

Page 4

important to stress that the deﬁnition of the horizontal distribution is invariant under the

transformation η7→ fη, since ker(fη) = ker(η), while the vertical distribution is not [33].

The vertical distribution is 1-dimensional and is described in terms of the so-called Reeb

vector ﬁeld, which is unique once a contact form is ﬁxed.

Proposition 1.1.Given a contact manifold (M, η)there exists a unique vector ﬁeld R,

the Reeb vector ﬁeld, such that

η(R)=1 and dη(R,·) = 0 .

Diﬀeomorphisms that preserve the contact structure are of particular interest. We

have seen that two contact forms deﬁne the same contact structure if they are equal up

to multiplication by a non-vanishing function: this allows to classify contactomorphisms

into two classes.

Definition 1.2.Denote (M, η)and (M′, η′)two contact manifolds which are diﬀeomor-

phic via ϕ:M→M′. We call the map ϕa

•contactmorphism, if there exists a non-vanishing λ∈C∞(M, R)such that ϕ∗η′=

λη,

•strict contactomorphism, if ϕ∗η′=η.

Similarly, we can also classify the corresponding inﬁnitesimal transformations.

Definition 1.3.Let (M, η)be a contact manifold. A vector ﬁeld Xon Mis called

•inﬁnitesimal contactomorphism, if LXη=τη for some τ∈C∞(M, R),

•strict inﬁnitesimal contactomorphism, if LXη= 0.

Finally, we want to give a coordinate description of a contact manifold. In Deﬁ-

nition 1.1 we have seen that the distribution is described by a 1-form. The following

theorem provides local canonical coordinates.

Theorem 1.1 (Darboux’s theorem).Consider a contact manifold (M, η)and a point

x∈M. Then there exist local coordinates (qi, pi, s)in a neighbourhood of xsuch that

the contact form is written as

η=ds −pidqi,

where here and in the following Einstein’s summation convention over repeated indices

will be always assumed.

The coordinates in Darboux’s theorem are called canonical coordinates. Note that in

these coordinates we have dη =dqi∧dpiand R=∂

∂s .

1.1 Contact hamiltonian flows and Jacobi structures

The contact form allows to deﬁne an isomorphism between functions and vector ﬁelds.

We call these vector ﬁelds contact gradients or contact Hamiltonian vector ﬁelds [33].

Page 5

Definition 1.4.Given a contact manifold (M, η)and a smooth function H:M→R,

the contact Hamiltonian vector ﬁeld associated with His the vector ﬁeld XH∈X(M)

such that

η(XH) = −H, dη(XH,·) = dH−dH(R)η. (3)

We denote the map that assigns to a function Hits associated contact Hamiltonian

vector ﬁeld as aη, so that XH=aη(H)and the function His called contact Hamiltonian

function.

Any contact Hamiltonian vector ﬁeld is an inﬁnitesimal contactomorphism, as shown

by a direct application of Cartan’s magic formula:

XHη=d(η(XH)) + dη(XH,·) = −R(H)η.

In particular, it is strict if and only if R(H) = 0. Remarkably, one can prove also the

opposite, i.e. any inﬁnitesimal contactomorphism is a contact Hamiltonian vector ﬁeld,

with the corresponding (contact) Hamiltonian function being recovered by using the ﬁrst

condition in (3).

In canonical coordinates (qi, pi, s), the contact Hamiltonian vector ﬁeld is given by [10,

11]

XH=∂H

∂pj∂

∂qj

+−∂H

∂qj−pj

∂H

∂s ∂

∂pj

+pi

∂H

∂pi−H

∂

∂s

and its integral curves satisfy the contact Hamiltonian equations of motion











˙qj=∂H

∂pj

˙pj=−∂H

∂qj−pj∂H

∂s

˙s=pi∂H

∂pi−H.

(4)

In what follows we will denote the contact Hamiltonian ﬂow of XH, or equivalently the

solution of the above system, by

ϕH

x0(t) := etXH(x0),

where x0∈Mis the initial condition.

Diﬀerently from its symplectic counterpart, a contact Hamiltonian function is not

conserved along the ﬂow of XH. Instead, its dynamics depends on R(H)as can be noted

in the following

Proposition 1.2.The time derivative of Halong the ﬂow of XHis

H=−R(H)H.

This means, in particular, that the surface deﬁned by H= 0 is invariant under the

contact Hamiltonian evolution and that this is generally not true for the other level

sets [10,11,30].

Classical Hamiltonian systems can be studied in terms of the algebra of Hamiltonian

vector ﬁelds on a symplectic manifold and in terms of the algebra of Hamiltonian functions

with the Poisson bracket. This duality persists also in contact Hamiltonian mechanics.

The set of contact Hamiltonian functions on a contact manifold (M, η)has a structure of

a (local) Lie algebra in the sense of Kirillov [27]: the bracket can be deﬁned by [14]

{f, g}η:= −η([Xf, Xg]) ,

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摘要：

TheflowmethodfortheBaker-Campbell-Hausdorffformula:exactresultsFedericoZadra∗1,AlessandroBravetti†2,AngelAlejandroGarcía-Chung‡3,4,andMarcelloSeri§11BernoulliInstituteforMathematics,ComputerScienceandArtificialIntelligence,UniversityofGroningen,Groningen,TheNetherlands2InstitutodeInvestigacionesenMa...

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