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Ph610 Analytical Mechanics Fall 2012 H.W. #5 -revised Due: 9/26/2012 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.,

< T >= −

12<

Fi'

i=1

N

∑ • ri > .

2. (40 points) Modified Kepler’s problem Consider a two-body system. (a) Derive the following orbital equation

dθ =dr

µr2 2µ

E −V (r) − 2

2µr2

⎛⎝⎜

⎞⎠⎟

; µ = reduced mass,=angular momentum

(b) Consider an inter-particle interaction, V (r) = −kr+hr2

, i.e. a perturbed gravitational

interaction. Find the exact solution r(θ) by integrating the equation in part (a).

(c) For E< 0 orbits, show that if − kr>>

hr2

then the orbit is a precessing ellipse with a

precession frequency ( Ω ) given by

Ω =

2πµhτ2

where τ is the orbital period for h=0.

(See GPS, Ch. 3 Exercise 21 for a discussion on using this as a model for precession of Mercury’s orbit)

(d) The term hr2

looks very much like the centrifugal barrier

2

2µr2, why does this term causes a

precession of the orbit while an addition to the centrifugal barrier through a change in does not cause a precession? 3. Coordinate Transformation (30 points) (a) In class, we introduced the metric tensor ( gij ) for a given set of coordinate axes (unit vectors), in particular, the metric tensor for a set of orthogonal unit vectors is gij = δ ij . Here you will work out explicitly the metric tensor for a set of non-orthogonal unit vectors in 2-D;

x1 = i , x2 =32i + 12j (In solid state physics, these vectors will generate a triangular lattice).

There is another set of unit vectors, b1, b2 which are orthogonal normal to x1, x2 , that is, bi • x j = δ ij . Find b1, b2 (in solid state physics, b1, b2 are the reciprocal lattice vectors). A vector can be expressed in terms of any basis vectors, i.e.,

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r = xi xi = bibi . Show that bi = gij x j . (b) 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. (c) 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 this position vector is given by r (t) = xi ' xi '(t) = xi (t)xi , xi ’s are the fixed space-axes. Let’s xi be a set of orthogonal axes and xi '(t) = Aij (t)x j , show that the coordinates are also given by the same transformation, xi

' = Aij (t)x j (t) . As an example, show explicitly for a rotation of the x-y plane about the z-axis through an angle θ . Now, Let’s xi be a set of non-orthogonal axes whose metric tensor is gij . Given xi '(t) = Aij (t)x j , find the transformation for the coordinates.