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md2011-01-note 8

# md2011-01-note 8 - If on the other hand we consider a...

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I If, on the other hand, we consider a system of atoms in Cartesian coordinates r i and define the kinetic energy – as usual – as K X i , α p 2 i , α 2 m i (4) where p i = m i d r i / d t , i runs over the particles and α over the coordinates ( x , y , z ) and m i denotes the mass of a particle i , then the equation of motion becomes m i ¨ r i = f i (5) where the force on atom i is (since r i K = 0) f i = r i L = - r i U (6) Notes

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I Generalized momentum is defined as p k = L ˙ q k (7) and the Hamiltonian equations of motion for the generalized coordinates are ˙ q k = H p k ˙ p k = - H q k (8) Notes
I As we saw before, the Hamiltonian is defined as H ( p , q ) K ( p ) + U ( q ) (9) I Since L K - U U = K - L and K i , α p 2 i , α 2 m i , H ( p , q ) = X k ˙ q k p k - L ( q , ˙ q ) (10) where ˙ q k is assumed to be a function of the momenta p , and U is assumed to be independent of velocities and time, U ( r N ) . Notes

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I For Cartesian coordinates ( x , y , z ), Hamilton’s equations become ˙ r i = p i m i (11) ˙ p i = - r i U = f i
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md2011-01-note 8 - If on the other hand we consider a...

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