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By Kapovich M.

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Though potentially confusing, ϕ2 ◦ f ◦ ϕ−1 1 is often called just ‘f ’, as long as the choice of coordinates is clear from the context. 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 .

If ω and ω are the angular velocity vectors defined with respect to the two coordinate systems, show that ω = P ω. 34 Lagrangian and Hamiltonian mechanics Fig. 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.

Note that the first equation is irrelevant. 24) is a conserved quantity. 43 Every Newtonian potential system, mi q ¨i = − ∂V , ∂qi i = 1, . . 26) for the Hamiltonian N H(q, p) := i=1 1 pi 2mi 2 + V (q) . 25) and p(t) = (p1 (t), . . , pN (t)) = (m1 q˙ 1 (t), . . e. p(t) is linear momentum. Note that for such a p(t), we have H(q, p) = N 1 ˙ i 2 + V (q) = K + V . 25) is equivalent to the following ˙ first-order system in variables (q, q), d qi = q˙ i , dt mi d ∂V q˙ i = − , dt ∂qi i = 1, . . , N .

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