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Ph610 Analytical Mechanics Fall 2009 H.W. #5 (updated) Due: 10/2/2009 HW#5: Multi-body problem-II, Coordinate transformation due to rotation 1. (30 points) Consider a system in which the total forces acting on the particles consist of ! ! conservative forces Fi ' and frictional forces fi . Show that if the frictional force is proportional to the particle’s velocity, then the frictional force does not contribute to the virial theorem , i.e., 1 N !' ! < T >= ! < " Fi • ri > . 2 i =1 2. (40 points) Consider a two-body system. (a) Derive the following orbital equation !dr d! = ; µ = reduced mass,!=angular momentum 2# !2 & µr 2 E " V (r) " µ%$ 2 µr 2 ( ' k h (b) Consider an inter-particle interaction, V (r) = ! + 2 , i.e. a perturbed gravitational r r interaction. Find the exact solution r(! ) by integrating the equation in part (a). k h (c) For E< 0 orbits, show that if ! >> 2 then the orbit is a precessing ellipse with a r r ! precession frequency ( ! ) given by ! 2"µh != where ! is the orbital period for h=0. # "2 h !2 (d) The term looks very much like the centrifugal barrier , why does this term causes a r2 2 µr 2 precession of the orbit while an addition to the centrifugal barrier through a change in ! does not cause a precession? (3) (30 points) (a) We have shown in class that a rotation of a set of orthogonal axes into another set of orthogonal axes is described by an orthogonal coordinate transformation, i.e., ! xi ' = Aij x j , where A ia an orthognal matrix, that is AA = I . Now consider a rotation of a set of non-orthogonal axes whose metric tensor is given gij , find the corresponding condition on the transformation matrix A. (b) Let’s go back to orthogonal axes. Pick a point on a rigid object and let its position vector be ! r = xi ' xi ' with respect to the body axes. Let the rigid body undergoes rotational motion. Hence ˆ ! this position vector is given by r (t) = xi ' xi '(t) = xi (t) xi . ˆ ˆ Let xi = Aij (t)x j (t) , show that xi '(t) = Aij (t) x j ' ˆ ˆ ! " ! ! ! (c) Extra credit (15 points) For a pure rotation, the velocity vector r (t) = ! (t) " r (t) , where ! (t) is the instantaneous rotational frequency of the object. Here is a brute force method to show this ! relationship in terms of the Euler’s angles. According to part b, r (t) = xi ' xi '(t) = xi ' Aij (t) x j , ˆ ˆ ! " " ! " hence r (t) = xi ' Aij (t) x j . Now express r (t) in terms of the instantaneous body axes, ˆ ! " " " # ˆ' r (t) = xi ' Aij (t) x j = xi ' Aij (t) A jk xk (t) . The transformation matrices in terms of Euler’s angles are ˆ given in GPS (4.46) and (4.47). !" ! ! ! Compute r (t) in the body axes and show that it can be written as ! (t) " r (t) . Express ! (t) in ! the body axes, i.e., ! (t) = ! i (t) xi' (t) and ! i (t) in terms of the Euler’s angles. ˆ ! For your convenience, A is given below % cos! cos " # cos$ sin " sin ! # sin ! cos " # cos$ sin " cos! sin $ sin " ( ' * ! A(t) = ' cos! sin " + cos$ cos " sin ! # sin ! sin " + cos$ cos " cos! # sin $ cos " * ' sin $ sin ! sin $ cos! cos$ * & ) is expressed in terms of the Euler’s angles which are function of time. *** The algebra is quite involved, you may use Mathematica or similar program. *** (If you don’t have such program, the text provided an alternative derivation of ! i (t) and the results are given in Eq. (4.87))