# Hamiltonian Systems Based on a Lie Algebra

# Hamiltonian Systems Based on a Lie Algebra

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is *not* invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.

*Keywords:*
Rigid body, Lie algebra, vector space, Virasoro algebra, KdV equation, Burgers equation, spectral method, vortex dynamics, Clebsch variables, hamiltonian formalism

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