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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.
- Spring '10