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Unformatted text preview: • Example. • One of the most important examples is the Lagrangian quadratic in velocities L ( q , ˙ q ) = 1 2 n X i , j = 1 g i j ( q )˙ q i ˙ q jV ( 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 1Form • The Poincar´e 1form λ is a 1form 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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 Spring '10
 Wong
 Geometry, Derivative, Hamiltonian mechanics, Fiber bundle, Leonhard Euler, phase space, Cotangent bundle

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