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The Lie algebra of classical mechanics

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GCS - Geometry, compatibility and structure preservation in computational differential equations

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket which is useful in geometric 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.  Therefore, we study and give a complete description of 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.
Joint work with Ander Murua.




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