Phys 325 Spring 2011 Lecture 21 - Physics 325 Lecture 21...

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Physics 325 Lecture 21 Hamiltonian Dynamics Because the equations of motion of a closed system of particles does not change in time, the Lagrangian for a closed system cannot have any explicit time dependence 0 L t The total time derivative of the Lagrangian is then jj j j qq dL L L L dt q t q t t LL      Using Lagrange’s equations, the first term in the square brackets can be replaced L d L q dt q giving j j j j dL d L L dt dt q q dL q dt q    Therefore, we arrive at a conservation law 0 j j j Lq dt q  or constant j j L L q H q   (21.1)
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What is the thing we have called H ? For a conservative system we have jj LT qq  Taking Cartesian coordinates as an example, 2 j TT mx x T xx This is true in any generalized coordinate system (see Sec. 7.8 in the text) as long as the transformation equations from Cartesian coordinates to the generalized coordinates are independent of time. Assuming this to be true (and U independent of j q ) , Equation (21.1) is then ( ) 2 ( ) T U T T U H    and H is the total energy. The function H , defined in Equation (21.1), is called the Hamiltonian of the system. We may write     , , , , k k j j k k j H q p t q p L q q t (21.2)
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys 325 Spring 2011 Lecture 21 - Physics 325 Lecture 21...

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