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

By James Montaldi, Tudor Ratiu

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

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If Ω is strongly nondegenerate, (P, Ω) is called a strong symplectic manifold. Recall that Ω is weakly, respectively strongly, nondegenerate if the smooth vector bundle map covering the identity : T P → T ∗ P given by v → v := Ω(v, ·) is injective, respectively bijective, on each fiber. If P is finite dimensional, there is no distinction between these concepts and nondegeneracy implies that P is even dimensional. 38 II A Crash Course in Geometric Mechanics If the manifold is a Banach space V and the two-form is constant on V , then (V, Ω) is called a symplectic Banach space.

Summarizing, for classical mechanical systems in Euclidean space describing particle motion, whose total energy is given by kinetic plus potential energy, we have shown that Newton’s equations are equivalent to: • Lagrange’s equations • Hamilton’s variational principle • Hamilton’s equations of motion • Hamilton’s equations in Poisson bracket formulation. In the course of these lectures we shall focus on each one of these four pictures and shall explain the geometric structure underlying them when the configuration space is a general manifold.

6 Given a C k Lagrangian L : T Q → R for k ≥ 2, there exists a ¨ → T ∗ Q, where unique C k−2 map EL(L) : Q ¨ := Q d2 q (0) ∈ T (T Q) q(t) is a C 2 curve in Q dt2 is a submanifold of T (T Q) (second order submanifold), and a unique C k−1 56 II A Crash Course in Geometric Mechanics one-form ΘL ∈ Ω1 (T Q), such that for all C 2 variations qε (t) (defined on a fixed t-interval) of q0 (t) := q(t), we have b dA[q(·)] · δq(·) = EL(L) a d2 q dt2 · δq dt + ΘL dq dt b δq a where δq(t) = d dε qε (t), δq(t) = ε=0 d dε ε=0 dqε (t) .

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