The Lie algebra of classical mechanics

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the `Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra $\mathcal X$, spanned by `modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $\mathcal X$. We calculate the dimensions $c_n$ of its homogeneous subspaces and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.8249\dots$, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.


Introduction
Simple mechanical systems are defined by pairs (Q, V ), where the configuration space Q is a real Riemannian manifold and the potential energy V is a smooth real function on Q. The phase space T * Q has a canonical Poisson bracket and a kinetic energy T : T * Q → R associated with the metric on Q. Given two distinguished functions, namely the kinetic and potential energies, one can ask what Lie algebra they generate under the Poisson bracket.
In this paper we study, not the Lie algebra generated by a particular V and T , but the Lie algebra defined by the whole class of simple mechanical systems. That is, one should think of the dimension of Q as being arbitrarily large, and the metric and potential energies also being arbitrary. This 'Lie algebra of classical mechanics' was introduced in [13]. However, that paper has an error that is corrected here.
Numerical integrators based on splitting and composition are widely used in applications including molecular, celestial, and accelerator dynamics [8,12]. The vector field X which is to be integrated is split as X = A + B, where A and B have the same properties (e.g. Hamiltonian) as X, but can be integrated exactly. The integrator is a composition of the form where ∆t is the time step and exp(tX) is the time-t flow of X. The Baker-Campbell-Hausdorff formula gives Z ∈ L(A, B), the free Lie algebra with two generators. Requiring Z = ∆t(A + B) + O((∆t) p+1 ) for some integer p > 1 gives a system of equations in the a i and b i which must be satisfied for the method to have order p. In the case of general A and B, then, at each order n = 1, . . . , p there are dim L n (A, B) such order conditions. Here L n (A, B) is the subspace of L(A, B) consisting of homogeneous elements of order n.
In this approach it is assumed that the only identities satisfied by Lie brackets of A and B are those due to antisymmetric and the Jacobi identity. This is reasonable if one wants the method to work for all A and B. However, in the case of simple mechanical systems, the Lie algebra is never free, regardless of T , V , or the dimension of the system. There are always identities satisfied by the Poisson brackets of the kinetic and potential energy. The simplest of these is is a function of q only. So V and {V, {V, T }} commute. Thus, it was realized early on [11] that in deriving high-order integrators as in Eq. (1) for simple mechanical systems, the order conditions corresponding to {V, {V, {V, T }}} and to all its higher Lie brackets can be dropped. This means that more efficient integrators can be designed for this class of systems. Much work has been done on this special case, both because of its intrinsic theoretical and practical importance, and because it allows such big improvements over the general case. For example, one can design special ('corrector' or 'processor') methods of the form ϕψϕ −1 [2], special methods for nearly-integrable systems such as the solar system [2,20], special methods involving exact evaluation of the forces associated with the 'modified potential' (Eq. 3) [3], and so on-see [12] for a survey. All of these studies rely on the structure of the Lie algebra generated by kinetic and potential energy. Bases for the homogeneous subspaces of this Lie algebra have been constructed for small orders [3,4,15], and McLachlan and Ryland [13] attempted to construct the entire Lie algebra. However, their construction is in error starting at order 13.
The paper is structured as follows. In Section 2.1, we define our object of study, the Lie algebra of classical mechanics. It is the universal Lie algebra with two generators equipped with a grading by degree. (In the motivating example, the homogeneous components are homogeneous polynomials in p, and the two generators have degree 2 (the kinetic energy) and degree 0 (the potential energy).) We explicitly construct this Lie algebra as the direct sum of an abelian and a free Lie algebra ( §2.2) and specify an explicit generalized Hall basis ( §2.3). In §2.4 we show that the motivating example has no further identities in general.
Sections 3 and 4 consider two special cases. The first is that the kinetic energy is equal to 1 2 p · p but the potential energy is arbitrary. In this case we conjecture, but are unable to prove, that there are no further identities. We provide some supporting evidence for the conjecture. The second, related, case is associated to the linear Schrödinger equation. We show that the universal Lie algebra generated by commutators of a potential and a Euclidean Laplacian, graded by degree of differential operators, is identical to the Euclidean (p · p) case.
Finally, in Section 5 we describe the Lie algebra of classical mechanics quantitatively, enumerating the dimensions of its homogeneous components and their asymptotic rate of growth. Recall that in the free case, dim L n (A, B) ∼ 1 n 2 n , so that the 'entropy' of L(A, B) is equal to 2. We calculate that the entropy of the Lie algebra of classical mechanics is 1.8249111600523655937 . . ., a fundamental constant associated to classical mechanics. For simple mechanical systems, every Lie bracket of T and V is a homogeneous polynomial in p. Furthermore, the degrees of these polynomials combine in a natural way. We therefore introduce the following class P of Lie algebras.  Note that this implies for all n. We call the grading of L its grading by degree.
We also need the concept of a Lie algebra which is free in a certain class.
Definition 2 Let F be a Lie algebra of class P generated by a set X. Then F is called a free Lie algebra in the class P over the set X, if for any Lie algebra R of class P, every mapping X → R that respects the degrees of the elements can be extended to a unique homomorphism F → R of Lie algebras of class P. We denote as L P (A, B) the free Lie algebra in the class P where A has degree 2 and B has degree 0, and we call it the Lie algebra of classical mechanics.
Consider the grading by degree of the free Lie algebra L(A, B) as follows: Given smooth functions T (p, q) = 1 2 M(q)(p, p) and V = V (q) (p, q ∈ R d ), let us denote by C ∞ pol (R 2d ) the Lie algebra under the Poisson bracket of smooth functions C ∞ (R 2d ) that depend polynomially on p. Then C ∞ pol (R 2d ) is of class P, with T of degree 2 and V of degree 0.
Definition 4 Given smooth functions T (p, q) = 1 2 M(q)(p, p) and V = V (q) (p, q ∈ R d ), is the unique homomorphism of Lie algebras of class P such that Φ T,V (A) = T and Φ T,V (B) = V .

Characterization of the Lie algebra of classical mechanics
Given a set Y, L(Y) denotes the free Lie algebra over the set Y. Given two disjoint sets Y 1 and Y 2 , the Lazard factorization of free Lie algebras [10] states that The generalized Hall basis [17] for L(Y) (constructed in §2.3) can be obtained by iteratively eliminating elements from the generating set. We construct L P (A, B) by successively applying a Lazard factorization to eliminate the elements of degree 0. We begin by first eliminating B, which introduces new elements of degree 0 into the generating set which are eliminated at the next stage of an iteration. The minimum order of these new elements of degree 0 increases with each step of the iteration, allowing a passage to the limit. This leads to the following characterization of the Lie algebra of classical mechanics. We postpone its proof to Subsection 2.3.

Theorem 1
The Lie algebra L P (A, B) of classical mechanics satisfies where span(X) is abelian, of degree 0, and is isomorphic as a vector space to the free commutative nonassociative algebra generated by B under the operation Specifically, with one basis element associated to each rooted full binary tree (i.e., each node has degree 0 or 2). Because of the A in (7), the order of an element with n Bs is 2n − 1.
Note that the Jacobi identity In the Lie algebra generated by specific T (p, q) = 1 2 M(q)(p, p) and V = V (q), that is, in the image by the homomorphism Φ T,V given in Definition 4, for each pair ( In the Euclidean case T = 1 2 p · p, we have

Basis for the Lie algebra of classical mechanics
Given a totally ordered set (Y, >), generalized Hall basis are totally ordered basis (H, >) for the free Lie algebra L(Y) satisfying the following conditions [17]: 1. Y is a totally ordered subset of the totally ordered set H, Observe that H n = {U ∈ H : order(U) = n}. Clearly, one can construct a totally ordered set H satisfying conditions 1 and 2 by inductively determining H n (for n ≥ 2) from condition 2 provided a total order on ∪ n−1 k=1 H k has been chosen, and then arbitrarily extending that order relation to ∪ n k=1 H k . However, that construction of H does not guarantee in general the fulfillment of condition 3.
If instead of arbitrarily extending the total ordering of ∪ n−1 k=1 H k to ∪ n k=1 H k , one imposes that U 1 < U 2 provided that order(U 1 ) < order(U 2 ), then condition 3 is automatically satisfied.
A different generalized Hall basis H for the free Lie algebra L(A, B) that will allow us to construct a basis for the Lie algebra L P (A, B) of classical mechanics can be constructed by imposing a different condition to the total order relation that also guarantees the fulfillment of condition 3 provided that condition 1 and 2 are satisfied: We require that U 1 < U 2 provided that either where degree(·) refers to the grading by degree of the free Lie algebra L(A, B) given in Definition 3. Obviously, there is much freedom in extending the total ordering of ∪ n−1 k=1 H k to ∪ n k=1 H k for each n ≥ 2 in the inductive construction of the generalized Hall basis H. In what follows, we assume that a particular choice has been made to give a total ordering to the elements with a common degree and order. For instance, such a total order can be defined as follows The elements of order up to six of the corresponding generalized Hall basis are displayed (sorted according to the total order in H) in Table 1.
A basis B for the Lie algebra of classical mechanics L P (A, B) can be constructed as a subset of such a generalized Hall basis H = ∪ n≥1 H n . Indeed, B = ∪ n≥1 B n where B 1 = {A, B} and for n ≥ 2,

Realization of the Lie algebra of classical mechanics
Given a particular mechanical system with potential energy V (q) and kinetic energy T (p, q) = 1/2 M(q)(p, p), the Lie algebra generated under the Poisson bracket by V and T will typically have linear dependencies than are not present in L P (A, B), that is, ker Φ T,V = 0. However, we will show in this section that there are no linear dependencies that are shared by all possible mechanical systems with arbitrary degrees of freedom, that is, the intersection of the kernels of all the homomorphisms Φ T,V corresponding to all smooth mechanical systems reduces to 0. Specifically, we will show in Theorem 2 below that L P (A, B) can be realized as the projective limit of the Lie algebras of a sequence of mechanical systems with polynomial Hamiltonian functions. This is achieved by constructing (along the lines of the standard proof of independence of elementary differentials) for each n specific kinetic and potential energy functions such that the Lie algebra they generate has no additional identities up to order n.
Let B = {U 1 , U 2 , U 3 , . . .} be a totally ordered basis for L P (A, B), as constructed in Subsection 2.3 (in particular, satisfying that U i < U j if degree(U i ) < degree(U j )), such that, U 1 = A, U 2 = B, and for each n ≥ 1, the elements of {U i : d n−1 < i ≤ d n } are of order n.
Let for each n ≥ 1 denote as L P (A, B; n) the subspace of L P (A, B) (of dimension d n ) spanned by {U i : i ≤ d n }. For each i ≥ 3, we will define a monomial W i (p, q) in the variables p 0 , p 1 , . . . , p dn , q 1 , . . . , q dn as follows.
We omit the proof of Proposition 1, which is rather technical, and very similar to the proof of Lemma 2.3 in [7].

The case of classical mechanical systems in Euclidean space
We now consider classical mechanical systems with kinetic energy T (p) = 1 2 p · p. From now on, given a smooth potential V : R d → R, we will denote by Φ V the homomorphism Φ T,V given in Definition 4 for T (p) = 1 2 p · p (p ∈ R d ). As in §2.4, given a particular Euclidean mechanical system with potential V (q), the Lie algebra Φ V (L P (A, B)) generated under the Poisson bracket by V and T will typically have linear dependencies than are not present in L P (A, B), that is, ker(Φ V ) = 0.
We conjecture that there are no relations other than those inherited from L P (A, B) (antisymmetry, Jacobi identity, and that the vanishing of Lie bracket of two elements of degree 0) that are shared by all possible smooth Euclidean mechanical systems with arbitrary dimensions. More precisely, We will next give some hints that we believe could be helpful for a possible proof of the conjecture.
Let T be the set of free trees with two types of vertices, thick (ø) and thin (·), such that vertices with more than one edge are thick. Then for each W ∈ L P (A, B), Φ V (W ) can be written as a linear combination with integer coefficients of certain functions (referred to as elementary Hamiltonians) associated to each tree in T . (These trees are an alternative representation of the "free RKN trees" of [6,15].) It is useful to think of each edge connecting a thick vertex to a thin vertex as a free-end edge; such a free-end edge is "ready" to be grafted to (fill its free end with) a thick vertex of another tree.
Given u ∈ T , order(u) is the sum of the number of free-end edges plus twice the number of thick vertices minus one. As for the degree: degree(·) = 2, and for u ∈ T \{·}, degree(u) is the number of free-end edges of u.
We next define a binary operation ⊳ on span(T ).

Definition 5 ⊳ : span(T ) ⊗ span(T ) → span(T )
is defined as follows: ø ⊳ · = oe, u ⊳ ø = 0, · ⊳ u = 0 for all u ∈ T . If u ∈ T is a tree with degree(u) = m and k ≥ 1 thick vertices, then (i) u ⊳ · is the sum of the k trees with k thick vertices and degree m + 1 obtained by adding a free-end edge to one thick vertex of u.
(ii) ø ⊳ u is the sum of the m trees with k + 1 thick vertices and degree m − 1 obtained by grafting one free-end edge of u to a new thick vertex.
(iii) Given two trees u, v ∈ T \{ø, ·}, let k be the number of thick vertices of u and let m = degree(v). Then u ⊳ v ∈ T is the sum of the km trees obtained by grafting a free-end edge of v to a thick vertex of u.

Proposition 2 The binary operation
) endows the vector space span(T ) with a Lie algebra structure of class P. For each V ∈ C ∞ (R d ), there exists a homomorphism of Lie algebras of class P such that Ψ V (·) = T and Ψ V (ø) = V . By the universal property of L P (A, B), The Lie algebra homomorphism Ψ V in the statement of Proposition 2 can be defined as follows.
can be defined as follows: Ψ V (ø) = V , Ψ V (·) = T , and if u ∈ T has m + 1 vertices, then where each of the factors H(j 1 , . . . , j m ) [i] is associated to a vertex in u. More precisely, let us label each vertex of u from 1 to m + 1 and each edge of u with a different dummy index j 1 , . . . , j m . If the ith vertex is thin and the ℓth edge is connected to some thick vertex, then H(j 1 , . . . , j m ) [i] = p ℓ . If the ith vertex is thick and has r edges labelled by ℓ 1 , . . . , ℓ r , then In Table 5, the images by Φ V : L P (A, B) → C ∞ pol (R 2d ) and Θ : L P (A, B) → span(T ) of elements W of a basis of L P (A, B) of order up to 6 are displayed. The images by Ψ V : span(T ) → C ∞ pol (R 2d ) can be determined from the identity Φ V = Ψ V • Θ. The following result was proved (using a different notation and terminology) by Calvo in [6].  (A, B) of order up to 6. Repeated indices are summed from 1 to d. Proof . It is sufficient to show that for each u ∈ T , there exists V ∈ C ∞ (R d ), with d the number of edges of u, such that, given u ′ ∈ T , the value of Ψ V (u ′ ) at p = (1, . . . , 1) and q = (0, . . . , 0) is non-zero if and only if u ′ = u. Such a function V can be constructed for each u ∈ T as follows: Label each thick vertex of u from 1 to m (m = d + 1 − degree(u)), and each edge from 1 to d. Then, V = W 1 + · · · + W d+1 , where W i = q ℓ 1 · · · q ℓr provided that the edges adjacent to the ith vertex are those labelled by ℓ 1 , . . . , ℓ r respectively.

Proposition 3 Consider all polynomial potentials
Conjecture 1 is now equivalent to the following To support our conjecture, we compare the dimensions of the homogeneous subspaces T n of elements of T of order n = 1, 2, 3, . . . , 20, namely 2,1,2,2,4,5,10,14,27,43,82,140,269,486,939,1765,3446,6652 with the dimensions of L n P (A, B) displayed in Table 5. They coincide up to n = 8, and obey dim L n P (A, B) < dim T n for n > 8. The dimensions of the subspaces of homogeneous order and degree of L P (A, B) and T (displayed in Table 3 and Table 4) are also compatible with the conjecture.
To further support our conjecture, we next show that the restriction of Θ to [X, A] (and thus also the restriction to X) is injective: Clearly, T 0 can be identified with the set of (uncoloured) free trees. Hence, for each X ∈ X, Θ(X) can be seen as a linear combination of free trees. On the other hand, T 1 can be identified with the set of uncoloured rooted trees, each tree u ∈ T 1 with m vertices being identified with a rooted tree with m − 1 vertices (the unique thin vertex in u indicating the location of the root). With that identification, the restriction of the binary operation ⊳ to T 1 corresponds to the grafting operation on rooted trees. Hence, according to [5], span(T 1 ) is the free pre-Lie algebra on one generator with respect to the operation ⊳. Now, it can be shown that, given X 1 , X 2 ∈ X, The injectivity of the restriction of Θ to [X, A] is then a consequence of the main result in [1].

The Lie algebra of the time-dependent Schrödinger equation
Application of the operator splitting method to the time-dependent linear Schrödinger equation in R d leads to the Lie algebra of endomorphisms of C ∞ (R d ) generated under the commutator bracket, by the Laplace operator ∇ 2 = d j=1 ∂ 2 ∂q 2 j and the multiplicative operator V (q). We will show that this Lie algebra is isomorphic to the Lie algebra of classical mechanics in Euclidean space considered in previous subsection. Consider the unique Lie algebra morphism from L(A, B) to End(C ∞ (R d )) that sends A to 1 2 ∇ 2 and B to V . We will say that an endomorphism of C ∞ (R d ) is of degree 0 if it is a multiplicative operator, and of degree n ≥ 1 if it is a differential operator of order n. For n ≥ 0, we denote as End(C ∞ (R d )) n the vector subspace of endomorphisms of degree n. One can check that the homomorphism L(A, B) → End(C ∞ (R d )) sends elements of degree 0 to endomorphisms of degree 0. In addition, its kernel includes the Lie ideal I generated by Lie-brackets of elements of L(A, B) of degree 0. This implies that there exists a unique Lie algebra homomorphism such that ϕ V (A) = 1 2 ∇ 2 and ϕ V (B) = V . However, ϕ V is not compatible with the grading by degree. Indeed, provided that U ∈ L P (A, B) has degree(U) ≥ 1, ϕ V (U) is in general a linear combination of endomorphism of degree ≤ degree(U).
We now consider the linear mapφ V : L P (A, B) → End(C ∞ (R d )) defined as follows: Given U ∈ L P (A, B) with degree(U) = n, we setφ V (U) as the projection to End(C ∞ (R d )) n of ϕ V . Clearly,φ V is compatible with the grading by degree. Actually, it is a homomorphism of Lie algebras of class P, as it happens to be essentially the same as the homomorphism Φ V : L P (A, B) → C ∞ pol (R 2d ) considered in previous subsection. More precisely, consider the natural injection ν : This in turn shows that, if Conjecture 2 holds true, then ker(ϕ V ) = 0.

The entropy of the Lie algebra of classical mechanics
Let z(t) be the generating function for the enumeration of X by order. Let a(t) be the generating function for the enumeration of binary trees. It obeys a(t) = t + 1 2 (a(t) 2 + a(t 2 )) = t + t 2 + t 3 + 2t 4 + 3t 5 + 6t 6 + · · · = ∞ n=1 a n t n Because the order of an element of X with n Bs is 2n − 1, we have z(t) = t −1 a(t 2 ).
For any set W with generating function w(t) = n>0 |{W ∈ W : order(W ) = n}|t n , the dimensions of the homogeneous components of the graded Lie algebra L(W) = n>0 L n (W) are given by [9,14] dim L n (W) = The generating function for the grading by order of {A} ∪ X is t + tz(t) = t + a(t 2 ). An application of (10) together with the dimensions of the abelian part from (9) gives the dimensions of L P (A, B) as listed in Table 5.
The asymptotic growth of a n was obtained by Wedderburn [19] using a method that we review briefly. He works in terms of g(t) := 1 − a(t), which from (8) obeys the functional equation g(t 2 ) = g(t) 2 + 2t.
Therefore, the number of modified potentials of odd order n is dim(span X) n = O((r −1/2 ) n ) = O(1.2416267680863184293 n ).
Since the dimensions of the free part dominate the abelian part, we have that the entropy of the Lie algebra of classical mechanics is 1/α = 1.8249111600523655937 . . . .