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hw1 - Department of Chemical Engineering University of...

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Department of Chemical Engineering University of California, Santa Barbara ChE 210C Spring 2008 Instructor: Glenn Fredrickson Homework #1 Homework Due: Friday, April 18, 2008 1. Show that the total linear momentum is conserved for a system of N classical particles with an interaction potential U that only depends on the relative distances between particles. 2. Show that the Liouville operator as defined in class for a single particle confined to move in one-dimension has an adjoint operator L = -L . Argue that if we redefine the operator according to i L g ≡ { g,H } , where i - 1, then L is Hermitian. Generalize the proof to N particles in three-dimensions. 3. Derive Hamilton’s equations for a particle moving in two-dimensions under a central potential U ( r ). [Use polar coordinates ( r,θ )]. Which equation illustrates the law of conservation of angular momentum? Is angular momentum conserved if the potential also depends on θ ? 4. By making the assumption that the potential energy of an
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