120A_PS1.v2 - Chemistry 120A Problem Set 1 (due January 27,...

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Chemistry 120A Problem Set 1 (due January 27, 2010) 3. In class we showed that for one dimension, dE/dt = 0 implies ˙ p = - dV/dx i.e., f = ma . Now consider many dimensions. Let x i denote the Cartesian coordi- nate for dimension i , i = 1 , 2 ,...,N . The total potential energy is V ( x 1 ,x 2 ,...,x N ), and the total kinetic energy is N X i =1 p 2 i / 2 m i , where m i is the mass associated with dimension i . (a) Show that ˙ p i = - ∂V/∂x i (b) For one particle in the three dimensions x, y and z, show that (a) implies ˙ p = -∇ V 4. Consider two particles with masses m 1 and m 2 , and positions r 1 and r 2 . It is often convenient to describe this system with the relative coordinates r = r 2 - r 1 and the center of mass position R = ( m 1 r 1 + m 2 r 2 ) /M , where M = m 1 + m 2 is the net mass. For a classical system, you can also consider the associated velocities v = ˙ r and v cm = ˙ R
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120A_PS1.v2 - Chemistry 120A Problem Set 1 (due January 27,...

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