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We present in modern language the contents of the famous note published by Henri Poincaré in 1901, in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra. We write these equations, called the Euler-Poincaré equations, in an intrinsic form, without any reference to a particular system of local coordinates, and prove that they can be conveniently expressed in terms of the Legendre and momentum maps. We discuss the use of the Euler-Poincaré equation for reduction (a procedure sometimes called Lagrangian reduction by modern authors), and compare this procedure with the well known Hamiltonian reduction procedure (formulated in modern terms in 1974 by J. Marsden and A. Weinstein). We explain how a break of the symmetry in the phase space produces the appearance of a semi-direct product of Lie groups.
For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable system using separating variables. The variables are chosen in such way that half of them parameterizes the Lagrangian manifold, which coincides with the Liouville torus of the integrable system. The obtained representation is indecomposable and non-exponentiated.
We present a numerical approach to the visualization of the light cones, and hence the causal structure, of exterior Schwarzschild space, taking advantage of the symmetries of Schwarzschild space and the conformal invariance of null geodesics.
In this paper, we investigate some properties for generalized $f$-harmonic and $f$-biharmonic maps between two Riemannian manifolds. In particular we present some new properties for the generalized stress $f$-energy tensor and the divergence of the generalized stress $f$-bienergy.
Here we develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrödinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.