By A. T. Fomenko

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In Geometric Mechanics, this exactly corresponds to the body and spatial representations. To be more explicit, let vg ∈ Tg G and consider ξ b = Tg Lg−1 vg and ξ s = Tg Rg−1 vg . The relationship between spatial and body velocities can be written in terms of the inﬁnitesimal version of the adjoint action of G on itself, which is called the adjoint action of the Lie group on its Lie algebra. 5. The adjoint action of G on g is deﬁned as the map Ad : G × g −→ g given by Ad(g, ξ) = Adg ξ = Tg−1 Lg (Te Rg−1 ξ).

A group G equipped with a manifold structure is said to be a Lie group if the product mapping · and the inverse mapping g −→ g −1 are both C ∞ -mappings. A Lie group H is said to be a Lie subgroup of a Lie group G if it is a submanifold of G and the inclusion mapping i : H → G is a group homomorphism. , Lg (h) = gh and Rg (h) = hg. This allows us to consider the adjoint action of G on G deﬁned by Ad : G × G −→ G (g, h) −→ Adg (h) = Lg Rg−1 h = ghg −1 . Roughly speaking, the adjoint action measures the non-commutativity of the multiplication of the Lie group: if G is Abelian, then the adjoint action Adg is simply the identity mapping on G.

And U ⊂ Q open) such that [1, 253], 1. e. d is R-linear and d(α∧β) = dα∧β+(−1)k α∧dβ, where α ∈ Ω k (U ) and β ∈ Ω l (U ). 2. df = p2 ◦ T f , for f ∈ C ∞ (U ), with p2 the canonical projection of T R ∼ = R × R onto the second factor. 3. d ◦ d = 0. 4. e. if U ⊂ V ⊂ Q are open, then d(α|U ) = (dα)|U , where α ∈ Ω k (V ). Let f : Q −→ N be a smooth mapping and ω ∈ Ω k (N ). Deﬁne the pullback f ∗ ω of ω by f as f ∗ ω(q)(v1 , . . , vk ) = ω(f (q))(Tq f (v1 ), . . , Tq f (vk )), where vi ∈ Tq Q.