# PS3 - = d r/dt is the velocity We contend that if this is...

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Phys601/F11/Problem Set 3 Due 09/26/07 (to be upgraded) From Goldstein et al Do Ch 1 #8, #9, #10 (use #9), #20 (possible typo in #20: should be V 2 ) 3.1H Prove the following identities. A is a constant vector, r is a vector with Cartesian components x k , the components of / r are / x k , and r = | r | = ( r.r ) 1/2 . Note that r / r = 1 , where 1 is the unit tensor with Cartesian components 1 jk = δ jk . a. ( r.A )/ r = A b. (r 2 /2)/ r = r c. (d/dt)(r 2 /2) = r.v if v = d r /dt d. (d/dt)f[ r (t),t] = v. f/ r + f/ t if v = d r /dt 3.2H Suppose a force, F , acting on a mass m has the form F = Q v. [ r D - D r ], where the order of the vectors in the dyadic is important, Q=constant, and D is a constant vector. Here, r (t) is the particle position vector, and v
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Unformatted text preview: = d r /dt is the velocity. We contend that if this is fundamental force of physics, the Newtonian equation of motion must be rewritable as 1 (d/dt)( ∂ L/ ∂ q k . ) = ( ∂ L/ ∂ q k ), where L = T – U, T is the kinetic energy, and U = U(q k . , q k, t). Here q k (t) are generalized coordinates and there are no constraints assumed in the formulation of F . (Since there are no constraints, one may let q k → r .) Show that F can be cast into Lagrangian formulation by finding a U( r , v ). There is a systematic way to do this but it can be done by trial and error. 1 Not all v-dependent forces can do this – for example, the friction force, -μ v , cannot....
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