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The Lie algebra of classical mechanics

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  • 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.

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  • 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
     | Show Table
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    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
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    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 $d$

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    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
     | Show Table
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    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
     | Show Table
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