University of Warwick, Department of Physics
Second Year Module: Hamiltonian Mechanics-PX 242 (2000)
Preface.
If you want best value out of these problems you will attempt them as the course unfolds. They should definitely not be regarded as just an exam-time revision aid.
Problems 1. This is a 'warm-up' exercise on minimisation which should help you with the start of the course.
1. Find the value of M which minimises when
(i) the constant a is positive
(ii) the constant a is negative.
[this is (part of) the Landau theory of ferromagnetism, where M is the magnetisation and a changes sign at the Curie temperature ]
2.
The potential energy of the system of (one dimensional) springs above is
.
Minimise this with respect to and .
3. The generalisation of question 2 from three to (N+1) springs gives potential energy
where and are the fixed ends. Show that minimising this energy with respect to any of the intermediate positions leads to the equation
,
i.e. equal extension of all the springs.
[ Hint Only the energy terms with n=i and n +1=i are relevant. ]
4. Consider the 'Action'
where and are fixed.
By substituting , where will have to be zero at both ends, verify that the 'straight line path' is the one which minimises A.
Notice the parallels between questions three and four.
Problems 2. Lagrangian Mechanics
5. Starting from , where ,
show that the condition for A to be stationary with respect to small (infinitesimal) variations in , arbitrary except that they preserve and , leads to the Euler-Lagrange equation
,
and keep re-doing this until you can do it without consulting your notes!
6. The harmonic oscillator has kinetic energy and potential energy . Confirm (using the previous question!) that leads to the correct equation of motion.
7. (see also question 10)A ball rolls down an inclined plane without sliding, as shown in the diagram. The moment of inertia of a sphere about its centre is given in terms of its mass and radius by . /
(a) Verify that the down-slope velocity of the ball is , and hence show that the total kinetic energy is .
(b) Show that the potential energy is given by
+ terms independent of .
(c) From the Lagrangian find the equation of motion. Hence show that the downslope (linear) acceleration of the ball is given by .
8. Consider the following Lagrangian:
Show that one Euler-Lagrange equation gives and find the other.
By eliminating from the equations (or otherwise) show that the motion of has a characteristic (angular) frequency . What physical system does the Lagrangian model?
9. A mass is free to swing in a plane, suspended on a spring approximated as ideal and of zero rest length. Show that the Lagrangian takes the form
,
interpreting the notation.
Find the equations of motion, and hence also the point of mechanical equilibrium.
For vertical oscillations about the equilibrium point, when q stays zero, show that the (angular) frequency of motion is w.
Consider small angular oscillations about the equilibrium point in the approximation that r stays fixed, and show that their frequency is very simply related to that of vertical ones. How valid is the fixed r approximation?
NB The corresponding experiment works well using a metal 'slinky' for the spring.
10. (an extension exercise, for the brave) Question 7 is rather more interesting when the angle of the slope is allowed to be an (externally imposed) function of time, .
(a) Show that the kinetic energy of the ball is now
,
where is the distance (along the slope) between the point of pivoting of the slope and the point of contact of the sphere.
(b) Find the equation of motion, and confirm that when (w =constant) this reduces to , which is relatively straightforward to integrate.
Problems 3 Conservation Laws
11. Motion in a Central Potential.
(a) Show that the kinetic energy of a simple classical particle of mass m is given, using plane polar co-ordinates, by .
(b) From the resulting Lagrangian, , find the momenta conjugate to respectively. Given that V depends only on r explain why = constant.
(c) Compute the Hamiltonian explicitly from and confirm that this gives the same as . What feature of the Lagrangian forces this to be a constant of the motion, ?
(d) Express H in terms of and not .
[Hint:: ].
Hence show that the shape of the 'orbit' obeys the equation
.
12. Gravitational Orbits
(a) Building on question 11, consider now the special case corresponding to orbits under Gravitational or attractive electrostatic forces. Show that the change of variable from r to as variable (which is more obvious to choose in two steps, via ), leads to the equation
constant.
By comparing this with the statement of conservation of energy for a simple harmonic oscillator, or otherwise, show that it leads to an equation for the shape of the orbit of the form
which is the classical equation for a circle (), an ellipse (), a parabola () and a hyperbola () respectively.
Find expressions for the parameters , which are known as the 'semi-latus rectum' and 'ellipticity' respectively.
[Question 12 continued]
(b) For enthusiasts.! Reconsider part (a) when . You should find a similar solution goes through but now , where , leading to 'precession of the perihelion'. The latter means that the furthest point of successive orbits is at a different angle each orbit. Observations of this phenomenon for the planet Mercury have been cited as evidence for departure from Newtonian Gravity, in favour of Einstein's General Relativity.
13. The Gyroscope.The Lagrangian of the gyroscope spinning and pivoting freely about the origin as sketched here, is given by
,
where the precise significance of the various constants can be inferred by considering simple special cases. /
(a) Find the momenta canonically conjugate to the corresponding angles. Explain why is not a constant of the motion, but the other two momenta are.
(b) Write down the Hamiltonian, and explain why it is conserved. By substituting in terms of show that
Problems 4 Normal Modes14 Starting from
for the system of masses and springs shown to the right, /
find the force matrix and hence show that the normal mode [angular] frequencies obey
and thus .
It is also useful to confirm that you get the same frequencies using as co-ordinates and , giving a simpler force matrix but a non trivial inertia matrix.
15. Consider a pendulum consisting of two equal point masses suspended by successive links of equal length and negligible mass, which (as discussed in lectures for a slightly different case) has kinetic and potential energies
, .
Show that the inertia matrix for small angular displacements away from equilibrium is given by and that the corresponding force matrix is .
Hence show that the normal mode frequencies obey , and find and describe the shape of the corresponding normal mode motions.
16. Symmetry considerations
The out-of-plane motion of a planar symmetric ring molecule of N atoms is modelled by the Lagrangian
where the potential energy represents bending energy and it is understood that for , .
(a) In the case corresponding to a square planar molecule, the force matrix resulting is . Find the eigenvectors by symmetry considerations first, and then check that they give normal mode frequencies (squared) of 0 (three times) and (once).
[Hint: no need to calculate any determinants - just check the equations of motion.]
Interpret physically the modes with zero frequency.
(b) Consider now the case whose force matrix (check!) has rows and cyclic permutations thereof. Symmetry under commuting reflections should lead you to the expectation of two even-even modes, two even-odd, one odd-even and one odd-odd (the first mirror being through two of the atoms, the second not). From your interpretation of the zero modes in part (a), you should be able to anticipate the nature of one even-even mode and one even-odd mode (as well as the odd-even mode). Then by orthogonality you can determine the shapes of the other even-even and even-odd modes, and given all the modes find all the frequencies.
(c) Is there an easier way to approach part (b)? Consider classifying modes according to eigenvalues under the symmetry of rotation through ......
Problems 5 Hamiltonian Formulation
17. (a) Find Hamilton's Equations of Motion for the one dimensional system with Hamiltonian ,
where is a constant (the particle charge).
(b) Exploit and one of your equations of motion to show that the corresponding Lagrangian is .
[Note: the three dimensional versions of these equations, with A and V depending on time as well as three dimensional position , is a complete statement of the motion of a charged (non-relativistic) particle in imposed electromagnetic fields. ]
18. Starting from the relativistic Lagrangian, , find the canonical momentum conjugate to x and hence show that Hamiltonian is
[In quantum mechanics this Hamiltonian gives interesting difficulties - you cannot ignore the negative possibility for the square root, leading Dirac to predict the existence of positrons; he was awarded the Nobel prize when they were observed soon afterwards.]
19.(a) Revisit motion in a central potential as in question 11, governed by the Lagrangian
.
Show (perhaps using your answer to 11(c)) that the corresponding Hamiltonian as a function of co-ordinates and canonically conjugate momenta is
Confirm from Hamilton's equations of motion that is a constant, and hence that the radial motion is that of a one dimensional particle in total potential
.
(Notice how the kinetic energy associated with the conserved angular momentum now appears as a 'Centrifugal Potential Energy'.) Sketch vs r for and . What sort of orbit do oscillations governed by this potential correspond to?
(b) Enthusiasts can pursue the same approach to the gyroscope Hamiltonian obtained in question 13(b). The effective potential as a function of angle in that problem now has a variety of shapes depending on the values of . Oscillations of correspond to 'nutation'.