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Chapter 1: Mechanics 7 the equations of Lagrange can be derived: d dt L ˙ q i = L q i When there are additional conditions applying to the variational problem δ J ( u )=0 of the type K ( u )= constant, the new problem becomes: δ J ( u ) - λδ K ( u . 1.7.2 Hamilton mechanics The Lagrangian is given by: L = T q i ) - V ( q i ) . The Hamiltonian is given by: H = ˙ q i p i - L . In 2 dimensions holds: L = T - U = 1 2 m r 2 + r 2 ˙ φ 2 ) - U ( r, φ ) . If the used coordinates are canonical the Hamilton equations are the equations of motion for the system: dq i dt = H p i ; dp i dt = - H q i Coordinates are canonical if the following holds: { q i ,q j } =0 , { p i ,p j } , { q i j } = δ ij where { , } is the Poisson bracket : { A,B } = ± i ² A q i B p i - A p i B q i ³ The Hamiltonian of a Harmonic oscillator is given by H ( x,p p 2 / 2 m + 1 2 m ω 2 x 2 . With new coordinates ( θ ,I ) , obtained by the canonical transformation x = ´ 2 I/m ω cos( θ ) and p = - 2 Im ω sin( θ ) , with inverse θ =arctan( - p/m ω x ) and I = p 2 / 2 m ω + 1 2 m
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.

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