Article Contents
Article Contents

# 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 $\mathfrak{X}$, spanned by 'modified' potential energies and isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $\mathfrak{X}$. 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., that the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.

Mathematics Subject Classification: Primary: 17B01, 70G45.

 Citation:

• Table 1.  Elements of order up to 6 of a generalized Hall basis $\mathcal H$ for $L(A,B)$. The elements in the Lie ideal $\mathcal{I}$ generated by $[L^0(A,B),L^0(A,B)]$ are depicted in red. The elements belonging to the basis $\mathcal B$ for $L_{\mathfrak P}(A,B) = L(A,B)/\mathcal{I}$ are depicted in black. The elements are listed in the total order defined in §2.3

 $U$ order$(U)$ degree$(U)$ $B$ 1 0 $[B,[B,A]]$ 3 0 ${[B,[B,[B,A]]]}$ 4 0 $[[B,[B,A]],[B,A]]$ 5 0 ${ [B,[B,[B,[B,A]]]]}$ 5 0 ${[[B,[B,[B,A]]],[B,A]]}$ 6 0 ${[B,[B,[B,[B,[B,A]]]]]}$ 6 0 $[B,A]$ 2 1 $[[B,[B,A]],A]$ 4 1 ${[[B,[B,[B,A]]],A]}$ 5 1 $[[[B,[B,A]],[B,A]],A ]$ 6 1 ${[[B,[B,[B,[B,A]]]],A]}$ 6 1 $[[B,A],[[B,[B,A]],A]]$ 6 1 $A$ 1 2 $[[B,A],A]$ 3 2 $[[[B,[B,A]],A],A]$ 5 2 $[[B,A],[[B,A],A]]$ 5 2 ${ [[[B,[B,[B,A]]],A],A]}$ 6 2 $[A,[[B,A],A]]$ 4 3 $[A,[[[B,[B,A]],A],A]]$ 6 3 $[A,[[B,A],[[B,A],A]]]$ 6 3 $[A,[A,[[B,A],A]]]$ 5 4 $[A,[A,[A,[[B,A],A]]]]$ 6 5

Table 5.  Dimensions of Lie algebras graded by order. Column 2: Of the free Lie algebra with two generators. Column 3: Of the Lie algebra of classical mechanics, $L_\mathfrak P(A,B)$. Column 4: Dimensions $x_n$ of the subspace of elements of degree 0 in $L^n_\mathfrak P(A,B)$. Column 5: Dimensions $y_n$ of the subspace of elements of degree 1 in $L^n_\mathfrak P(A,B)$

 $n$ $\dim L^n(A,B)$ $\dim L^n_\mathfrak{P}(A,B)$ $x_n$ $y_n$ 1 2 2 1 2 1 1 1 3 2 2 1 4 3 2 2 5 6 4 1 6 9 5 4 7 18 10 2 8 30 14 9 9 56 25 3 10 99 39 18 11 186 69 6 12 335 110 41 13 630 193 11 14 1161 320 88 15 2182 555 23 16 4080 938 198 17 7710 1630 46 18 14532 2786 441 19 27594 4852 98 20 52377 8370

Table 2.  Image by $\Theta$ and $\Phi_V$ of Prop.2 of elements in $L_\mathfrak{P}(A,B)$ of order up to 6. Repeated indices are summed from 1 to $d$

Table 3.  Dimensions of homogeneous subspaces of $L_\mathfrak P(A,B)$ of order $n$ (rows $n = 1,\ldots,18$) and degree $m-1$ (columns $m = 1,\ldots,18$)

 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 4 0 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 6 0 3 0 1 0 0 0 0 0 0 0 0 0 0 3 0 9 0 8 0 4 0 1 0 0 0 0 0 0 0 0 0 0 9 0 14 0 11 0 4 0 1 0 0 0 0 0 0 0 0 6 0 20 0 23 0 14 0 5 0 1 0 0 0 0 0 0 0 0 18 0 37 0 32 0 17 0 5 0 1 0 0 0 0 0 0 11 0 46 0 62 0 46 0 21 0 6 0 1 0 0 0 0 0 0 41 0 90 0 97 0 60 0 25 0 6 0 1 0 0 0 0 23 0 106 0 165 0 144 0 80 0 29 0 7 0 1 0 0 0 0 88 0 228 0 274 0 206 0 100 0 34 0 7 0 1 0 0 46 0 248 0 438 0 438 0 285 0 127 0 39 0 8 0 1 0 0 198 0 562 0 777 0 658 0 384 0 154 0 44 0 8 0 1

Table 4.  Dimensions of homogeneous subspaces of $\mathcal T$ of order $n$ (rows $n = 1,\ldots,18$) and degree $m-1$ (columns $m = 1,\ldots,18$)

 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 4 0 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 6 0 3 0 1 0 0 0 0 0 0 0 0 0 0 3 0 10 0 9 0 4 0 1 0 0 0 0 0 0 0 0 0 0 9 0 17 0 12 0 4 0 1 0 0 0 0 0 0 0 0 6 0 24 0 30 0 16 0 5 0 1 0 0 0 0 0 0 0 0 20 0 50 0 44 0 20 0 5 0 1 0 0 0 0 0 0 11 0 63 0 96 0 67 0 25 0 6 0 1 0 0 0 0 0 0 48 0 146 0 164 0 91 0 30 0 6 0 1 0 0 0 0 23 0 164 0 315 0 267 0 126 0 36 0 7 0 1 0 0 0 0 115 0 437 0 592 0 408 0 163 0 42 0 7 0 1 0 0 47 0 444 0 1022 0 1059 0 603 0 213 0 49 0 8 0 1 0 0 286 0 1300 0 2126 0 1754 0 856 0 265 0 56 0 8 0 1
•  [1] N. Bergeron and J. L. Loday, The symmetric operation in a free pre-Lie algebra is magmatic, Proc. Amer. Math. Soc., 139 (2011), 1585-1597.  doi: 10.1090/S0002-9939-2010-10813-4. [2] S. Blanes, F. Casas and J. Ros, Processing symplectic methods for near-integrable Hamiltonian systems, Celest. Mech. Dyn. Astr., 77 (2000), 17-35.  doi: 10.1023/A:1008311025472. [3] S. Blanes, F. Casas and J. Ros, High-order Runge-Kutta-Nystrom geometric methods with processing, Appl. Numer. Math., 39 (2001), 245-259.  doi: 10.1016/S0168-9274(00)00035-0. [4] S. Blanes and P. C. Moan, Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142 (2002), 313-330.  doi: 10.1016/S0377-0427(01)00492-7. [5] M. P. Calvo Cabrero, Métodos Runge-Kutta-Nyström Simplécticos, Tesis Doctoral, Universidad de Valladolid, 1992. [6] F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, International Math. Research Notices, 8 (2001), 395-408.  doi: 10.1155/S1073792801000198. [7] M. Grayson and R. Grossman, Models for nilpotent free Lie algebras, J. Algebra, 135 (1990), 177-191.  doi: 10.1016/0021-8693(90)90156-I. [8] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7. [9] S.-J. Kang and M.-H. Kim, Free Lie algebras, generalized Witt formula, and the denominator identity, J. Algebra, 183 (1996), 560-594.  doi: 10.1006/jabr.1996.0233. [10] M. Lazard, Groupes, Anneaux de Lie et problème de Burnside, Groups, Lie rings and cohomology theory, C.I.M.E. Summer Sch., Springer, Heidelberg, 20 (2011), 123-184.  doi: 10.1007/978-3-642-10937-9_2. [11] M. Markl and E. Remm, Algebras with one operation including Poisson and other Lie-admissible algebras, J. Algebra, 299 (2006), 171-189.  doi: 10.1016/j.jalgebra.2005.09.018. [12] R. I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput., 16 (1995), 151-168.  doi: 10.1137/0916010. [13] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434.  doi: 10.1017/S0962492902000053. [14] R. I. McLachlan and B. Ryland, The algebraic entropy of classical mechanics, J. Math. Phys., 44 (2003), 3071-3087.  doi: 10.1063/1.1576904. [15] H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, R. Soc. Lond. Philos. Trans. A Math. Phys. Eng. Sci., 357 (1999), 957-981.  doi: 10.1098/rsta.1999.0361. [16] A. Murua, Formal series and numerical integrators, Part I: Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251.  doi: 10.1016/S0168-9274(98)00064-6. [17] R. Otter, The number of trees, Ann. Math., 49 (1948), 583-599.  doi: 10.2307/1969046. [18] C. Reutenauer, Free Lie Algebras, London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. [19] N. J. A. Sloane, The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., 50 (2003), 912–915, https://oeis.org. [20] J. H. M. Wedderburn, The functional equation $g(x^2) = 2\alpha x + [g(x)]^2$, Ann. Math., 24 (1922), 121-140.  doi: 10.2307/1967710. [21] J. Wisdom, M. Holman and J. Touma, Symplectic correctors, Integration Algorithms and Classical Mechanics, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 10 (1996), 217-244. [22] H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.  doi: 10.1016/0375-9601(90)90092-3.

Tables(5)