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 X 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 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 X i = 1 L q i ˙ q i - L ( q , ˙ 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 Poincare 1-Form The Poincare 1-form is a 1-form on the cotangent bundle T * M dened in local coordinates ( q , p ) on T * M by = n X i = 1 p i dq i . Remarks. di geom.tex; April 12, 2006; 17:59; p. 45...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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