Geometric mechanics and symmetry: From finite to infinite by Darryl D. Holm

By Darryl D. Holm

Classical mechanics, one of many oldest branches of technology, has passed through a protracted evolution, constructing hand in hand with many components of arithmetic, together with calculus, differential geometry, and the speculation of Lie teams and Lie algebras. the trendy formulations of Lagrangian and Hamiltonian mechanics, within the coordinate-free language of differential geometry, are based and normal. they supply a unifying framework for plenty of possible disparate actual structures, similar to n-particle structures, inflexible our bodies, fluids and different continua, and electromagnetic and quantum structures. Geometric Mechanics and Symmetry is a pleasant and fast moving creation to the geometric method of classical mechanics, appropriate for a one- or - semester path for starting graduate scholars or complex undergraduates. It fills a niche among conventional classical mechanics texts and complicated glossy mathematical remedies of the topic. After a precis of the required components of calculus on delicate manifolds and easy Lie crew idea, the most physique of the textual content considers how symmetry aid of Hamilton's precept permits one to derive and examine the Euler-Poincare equations for dynamics on Lie teams. extra subject matters take care of inflexible and pseudo-rigid our bodies, the heavy best, shallow water waves, geophysical fluid dynamics and computational anatomy. The textual content ends with a dialogue of the semidirect-product Euler-Poincare relief theorem for perfect fluid dynamics. numerous examples and figures illustrate the cloth, whereas the various workouts, either solved and unsolved, make the booklet a worthwhile classification textual content.

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2 Rigid body motion takes place on a body angular momentum sphere along the intersections of level sets of the energy. 13 Consider an ellipsoid with axes coinciding with the x, y, z-axes, and suppose for simplicity that it is stationary for all time, so x = X. Show that, if the ellipsoid has semi-axes of lengths a, b, c, then its moment of inertia tensor is M (b2 + c2 ) 4 0 0 21 5 0 2 1 M (a + c2 ) 5 0 3 0 5. 14 Show that if a body is two- or three-dimensional, meaning that there is no line that contains all of the mass of the body, then its moment of inertia matrix is always invertible.

The usual definitions of ‘differentiable’ and ‘smooth’ do not apply to general maps f between manifolds, since the domain of f need not be an open subset of Euclidean space. One way to remedy this is to use local representations ϕ2 ◦ f ◦ ϕ−1 1 , because their domains and codomains are always open subsets of Euclidean spaces, so the usual definitions do apply. 42 Let M be a submanifold of Rp and N a submanifold of Rs . The map f is differentiable at a ∈ M if for every coordinate chart ϕ1 of M with domain containing a, and every coordinate chart ϕ2 of N with domain containing f (a), the composition ϕ2 ◦ f ◦ ϕ−1 1 is differentiable at ϕ1 (a).

A general definition of ‘graph’ is awkward for notational reasons, but the idea should be clear from the preceding examples. A submanifold is the union of one or more graphs of functions. The following definition ensures that these graphs fit together smoothly. Recall that a neighbourhood of a point a ∈ Rn is an open set U containing a. 1 An m-dimensional (embedded) submanifold of Rn is a subset M of Rn such that, for every a ∈ M , there is a neighbourhood U of a such that M ∩ U is the graph of some smooth function (with an open domain)1 expressing (n − m) of the standard coordinates in terms of the other m.

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