Download e-book for iPad: Geometric Mechanics and Symmetry: The Peyresq Lectures by James Montaldi, Tudor Ratiu

By James Montaldi, Tudor Ratiu

ISBN-10: 0521539579

ISBN-13: 9780521539579

Geometric mechanics borders natural and utilized arithmetic and comprises such disciplines as differential geometry, Hamiltonian mechanics and integrable structures. The resource of this assortment is the summer time university on Geometric Mechanics and Symmetry equipped via James Montaldi and Tudor Ratiu. Written with major enter from the contributors on the convention, those lecture notes are geared in the direction of pleasing the wishes of graduate scholars via their consciousness to aspect.

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Extra info for Geometric Mechanics and Symmetry: The Peyresq Lectures

Example text

We shall use this formula in the proof below. 2) is equivalent to φ∗t £X Ω = 0 ⇐⇒ £X Ω = 0 since φ∗t is an isomorphism. The name “locally Hamiltonian vector field” is justified by the following fact. Since £X Ω = iX dΩ + diX Ω and dΩ = 0, the condition £X Ω = 0 is equivalent to diX Ω = 0, that is, iX Ω is closed. The Poincar´e lemma implies the existence of a local function H such that iX Ω = dH. 6 (Energy conservation) Let φt be the flow of the Hamiltonian vector field XH . Then H ◦ φt = H. Proof Since d φt (z) = XH (φt (z)) dt II.

Ratiu, R. Tudoran, L. Sbano, E. Sousa Dias & G. Terra Notes of the courses given by Tudor Ratiu 1. Introduction . . . . . . 2. Hamiltonian Formalism . . . . 3. Lagrangian Formalism . . . . 4. Poisson Manifolds . . . . . 5. Momentum Maps . . . . . 6. Lie-Poisson and Euler-Poincar´e Reduction 7. Symplectic Reduction . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 37 47 59 78 94 121 154 1 Introduction These lecture notes are the direct result of presentations held in two consecutive years (2000 and 2001) at the Peyresq Conference Center, in the Alpes de Haute Provence, North of Nice. These summer schools were organized in conjunction with the Research Training Network MASIE (Mechanics and Symmetry in Europe) of the Fifth Framework Program of the European Commission and were intended for graduate students and postdocs who needed a crash course in geometric mechanics.

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Geometric Mechanics and Symmetry: The Peyresq Lectures by James Montaldi, Tudor Ratiu

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