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Unformatted text preview: Chapter 5 Hamiltonian Mechanics 5.1 The Hamiltonian Recall that L = L ( q, q,t ), and p = L q . (5.1) The Hamiltonian, H ( q,p ) is obtained by a Legendre transformation, H ( q,p ) = n summationdisplay =1 p q L . (5.2) Note that dH = n summationdisplay =1 parenleftbigg p d q + q dp L q dq L q d q parenrightbigg L t dt = n summationdisplay =1 parenleftbigg q dp L q dq parenrightbigg L t dt . (5.3) Thus, we obtain Hamiltons equations of motion, H p = q , H q = L q = p (5.4) and dH dt = H t = L t . (5.5) Some remarks: As an example, consider a particle moving in three dimensions, described by spherical polar coordinates ( r,, ). Then L = 1 2 m ( r 2 + r 2 2 + r 2 sin 2 2 ) U ( r,, ) . (5.6) 1 2 CHAPTER 5. HAMILTONIAN MECHANICS We have p r = L r = m r , p = L = mr 2 , p = L = mr 2 sin 2 , (5.7) and thus H = p r r + p + p L = p 2 r 2 m + p 2 2 mr 2 + p 2 2 mr 2 sin 2 + U ( r,, ) . (5.8) Note that H is timeindependent, hence H t = dH dt = 0, and therefore H is a constant of the motion. In order to obtain H ( q,p ) we must invert the relation p = L q = p ( q, q ) to obtain q ( q,p ). This is possible if the Hessian, p q = 2 L q q (5.9) is nonsingular. This is the content of the inverse function theorem of multivariable calculus. Define the rank 2 n vector, , by its components, i = braceleftBigg q i if 1 i n p i n if n i 2 n . (5.10) Then we may write Hamiltons equations compactly as i = J ij H j , (5.11) where J = parenleftBigg O n n I n n I n n O n n parenrightBigg (5.12) is a rank 2 n matrix. Note that J t = J , i.e. J is antisymmetric, and that J 2 = I 2 n 2 n . We shall utilize this symplectic structure to Hamiltons equations shortly. 5.2. MODIFIED HAMILTONS PRINCIPLE 3 5.2 Modified Hamiltons Principle We have that 0 = t b integraldisplay t a dtL = t b integraldisplay t a dt ( p q H ) (5.13) = t b integraldisplay t a dt braceleftbigg p q + q p H q q H p p bracerightbigg = t b integraldisplay t a dt braceleftBigg parenleftbigg p + H q parenrightbigg q + parenleftbigg q H p parenrightbigg p bracerightBigg + ( p q ) vextendsingle vextendsingle vextendsingle t b t a , assuming q ( t a ) = q ( t b ) = 0. Setting the coefficients of q and p to zero, we recover Hamiltons equations....
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 Fall '11
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 mechanics

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