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differential geometry w notes from teacher_Part_21

differential geometry w notes from teacher_Part_21 - •...

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2.3. THE COTANGENT BUNDLE 41 Therefore, ( q 1 , . . . , q n , ˙ q 1 , . . . , ˙ q n ) give local coordinates for the tangent bun- dle T M . Let L : T M R be a map. Then L ( q , ˙ q ) is called the Lagrangian . The generalized momenta p i are defined by p i = L ˙ q i . The momenta are functions on T M , that is, p : T M R . Under a change of local coordinates q α = q α ( q β ) the momenta transform as components of a covector p α j = n i = 1 q i β q j α p β i , The matrix H ik ( q , ˙ q ) = 2 L ˙ q i ˙ q k is called the Hessian. Suppose that the Hessian is non-degerate det H ik 0 . Then the velocities can be expressed in terms of momenta ˙ q i = ˙ q i ( q , p ) , that is, there is a map ˙ q : T * M R . More generally, there is a map T M T * M . Thus, ( q 1 , . . . , q n , p 1 , . . . , p n ) give local coordinates for the cotangent bundle T * M . The cotangent bundle is called the phase space in dynamics. di ff geom.tex; April 12, 2006; 17:59; p. 44
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42 CHAPTER 2. TENSORS The Hamiltonian is a smooth function on the cotangent bundle H : T * M R defined by H ( q , p ) = n i = 1 L q i ˙ q i - L ( q
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Unformatted text preview: • Example. • One of the most important examples is the Lagrangian quadratic in veloci-ties L ( q , ˙ q ) = 1 2 n X i , j = 1 g i j ( q )˙ q i ˙ q j-V ( q ) , where g i j is a Riemannian metric on M and V is a smooth function on M . • Then the Hessian is g ik = ∂ 2 L ∂ ˙ q i ∂ ˙ q k and, therefore, nondegenerate. • The relation between momenta and the velocities is p i = n X j = 1 g i j ( q )˙ q j , ˙ q i = n X j = 1 g i j ( q ) p j . • The Hamiltonian is given by H ( q , p ) = 1 2 n X i , j = 1 g i j ( q ) p i p j + V ( q ) , 2.3.3 The Poincar´e 1-Form • The Poincar´e 1-form λ is a 1-form on the cotangent bundle T * M defined in local coordinates ( q , p ) on T * M by λ = n X i = 1 p i dq i . • Remarks. di ff geom.tex; April 12, 2006; 17:59; p. 45...
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