Classical Mechanics (PHY401)

Problem Set 1 Semester I, 2012

Part 1: Mathematical Preliminaries Matrices: 1. For two matrices A and B, show that

!+ ! !− ! = !! − !! if and only if A and B commute. 2. A is a diagonal matrix, B is some other matrix. If A and B commutes, B is also diagonal. 3. Show that the trace of a matrix remains invariant under orthogonal similarity transformations. 4. Show that the property of antisymmetry is invariant under orthogonal similarity transformations. 5. ! =  !!"! Show if H is hermitian, then U is unitary. 6. For ! =  !!,

a) Show !!! =  !!!. b) Show !!"!!! = !"!!!. c) If A is orthogonal find the condition on B.

7. Find conditions under which products of two hermitian matrices is also hermitian. Rotations: 8. Consider passive rotations in 2 dimensions. Show that they form a group. 9. Consider passive rotations about x and y axes, Rx(θ1) and Ry(θ2) respectively. Show that they do not commute. Take the limit θ1, θ2  → 0, and recheck the commutativity. 10. Show that !.! is a scalar under coordinate rotations. 11. If !×! = !, show that after a passive clockwise rotation about the z axis by an angle θ,  !′! = !! !"# ! +  !! !"# !. 12. For a non-Euclidean space the norm of a vector is defined as !! = −!!! + !!!. Show that the ‘rotation’ of the form:

! ! = !"#ℎ ! − !"#ℎ !− !"#ℎ ! !"#ℎ !

would preserve the norm for such a spatial geometry. Give example of such a transformation. Levi Civita: 13. Consider a magnetic dipole with magnetic moment ! in a magnetic field !, in absence of currents, then the force is given as ! =  ∇× !  ×! .  Use Levi Civita to show ! =  ∇× !  ×! =  ∇ ! ∙  ! . Assume ! constant. 14. Using Levi Civita show ! ∙ !!  ×!! = 0. 15. Using Levi Civita show

!× !×! = ! ∙ !  ! − ! ∙ !  ! − ! ! ∙  ! + !(! ∙  !) 16. Calculate !×[! ! !], where ! is the position vector and ! ! is a function of the radial coordinate only. 17. Is !× !×! the same as (!×!)×! ? 18. Find !×! ∙ !×! . 19. Use antisymmetry of !!"#to show ! ∙ !×! = 0. 20. Show !!"#!!!" = 6. Pauli Matrices: 21. The Pauli spin matrices are:

!! =0 11 0 ,      !! =

0 −!! 0 ,        !! =

1 00 −1 .

Show: a) (!!)! = !! b) !!!! = !!!"#!! c) !!!! + !!!! = 2!!"!!

22. Using the commutators and anticommutators of Pauli spin matrices show:

! ∙ ! ! ∙ ! = ! ∙ !  !! + !! ∙ (!×!) 23. Verify that for a rotation about x axis, the corresponding Q – Matrix is given by:

! = cos!2 !! + !!! sin

!2  

24. Use above relations to argue why infinitesimal rotations always commute.

Part 2: Lagrangian Dynamics and Application to Central Force Field: Lagrangian Dynamics: 25. Show that Lagrange’s equations are invariant under a point transformation of generalized coordinates defined as = 1 2( , ,.., , )i i nq q s s s t

Hint: Show ⎛ ⎞∂ ∂

− =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠&0

i i

d L Ldt s s

given that ⎛ ⎞∂ ∂

− =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠&0

i i

d L Ldt q q

for all i.

26. Prove that ( , )' df q tL Ldt

= + also satisfies the Lagrange’s equation satisfied by L.

27. Prove Nelson Form of Lagrange’s equation jjj

QqT

qT

=∂∂

−∂∂ 2

28. a) A particle of mass m is moving in a central potential, write down the Lagrangian and Lagrange’s Equation. Point out the cyclic coordinate and the conjugate momentum? Show that the differential equation for orbit is given by

+ = −2

2 2 2

( )d u mf uudθ u l

Also find out the effective potential and draw the energy diagram for the inverse square attractive force.

b) A simple pendulum consisting of a mass 2m at one end and a mass 1m at the point of support that moves in a vertical plane. Find out the equation of motion using the Lagrangian. Comment on the limitm1!" . c) Consider a coplanar double pendulum in which mass !! hangs from a fixed point by a massless string of length !! and a mass !! hangs from !!. Find the frequencies of small oscillations. What happens in the limit !! !! → ∞       d) A particle initially at rest starts falling from a height under the influence of gravity and air-resistance, which is given by zk− ; derive the Lagarange’s equation using appropriate Rayleigh dissipative function. At what rate energy is dissipated? Find the terminal velocity.

e) Using Lagrangian find out the equation of a simple pendulum. Show that when                        amplitude  (a)  is  not  very  small,  the  time-­‐period  of  oscillation  

= + +2

42 ( ),16

l aT π O ag

29. Lagrange’s equation for system with non-holonomic constraint: (Refer to the relevant discussion from Goldstein, Chapter 2, given the differential form of constraint equation use Lagrange’s equation of first kind.) a) A cylinder rolling down an inclined plane with no slipping; find out the                      velocity  at the bottom of the inclined plane.                  b) Particle moving on the surface of a sphere under the influence of gravity if   starts from the top 0=θ . At what angle it will fly off the sphere? 30. Show that if the potential in the Lagrangian contains velocity dependent terms, the canonical momentum conjugate to a coordinate denoting rotation !  of the entire system is no longer the angular momentum !! but is given by

= − ⋅ ×∇∑r r

iθ θ i ri

p L n r U

in the gradient operator, the derivative is with respect to the velocity components and n̂ is the unit vector of rotation. Find out this conjugate momentum when particles with !! is moving in a magnetic field. Central Field Motion: 31. Find out the condition for the stability of circular orbits in a force field of potential ! ! =  − !

!!"#(− !

!).

32. Determine the geometric trajectory !(!) for a particle of mass m moving in a central potential    ! ! = − !

!.

a) Solve the problem by solving the orbit equation given in terms of ! = 1 !  . (see                    previous  problem  set)     b) Using the equivalent one-dimensional Hamiltonian derive the equation

! = !! +!2!

!"  1

!!× ! − !!""  

and use this to derive the orbit. (These two problems are solved in details in Goldstein and other books; you must study it from them). 33. A particle of mass m is moving in a central potential ! ! = !

!!!!; use equation

given in 2(b) to find out the trajectory. 34. Suppose there is a small correction to Newton’s laws of gravitation so that the potential energy of a two-particle system is given by ! ! =  − !"#

!+   !

!!.

Show that the shape of the orbit is described by

( )⎡ ⎤⎛ ⎞⎢ ⎥= + + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

1/2

02

21 cos 1a mαe θ θr l

(We have assumed that M>>m.)  

35. a) Show that for a particle moving in an inverse square force field the Runge-Lenz vector is a constant of motion (Solved in Goldsetin) b) Consider a charged particle of mass m and charged q moving in the presence of

an electromagnetic field of the form: !E =Q

!rr3

and =rr3

rB Mr

show that the

angular momentum is not conserved. Show that the angular momentum! = !(!×!) is not constant but the

vector !D =!L ! qM

!rr

is. (We are using Gaussian Unit with c=1.)

36. A particle of mass m is moving in a circle of radius R in the presence of a central force whose origin lies on the circumference of a circle. Find the form of the potential. What is the total energy of the particle assuming V (r)! 0 as r!" . Find out the time period of motion as a function of the orbital angular momentum, R and m. Many problems those were discussed in the class or given as homework problems are not written explicitly. You must NOT forget them.