Faculteit Technische Natuurkunde - TU Eindhoven
Tentamen Theoretische Klassieke Mechanica (3EMX0)
Maandag 14 april 2014, 09.00–12.00.
Het tentamen levert max. 100 punten op, waarvan de verdeling hieronder is aangegeven.
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(1) Korte vraagstukken
Geef bij jouw antwoord of uitwerking een korte argumentatie. Bij een ernstige fout in de argumentatie worden geen punten toegekend.
a) Wat zijn ‘gegeneraliseerde co¨ordinaten’ ~ ? Geef voorbeelden.
b) Wat is het verschil tussen faseruimte en configuratieruimte?
c) Wat is de essentie van het principe van d’Alembert.
d) Leg uit wat een inertiaalstelsel is.
e) Van welke basisveronderstellingen is Newton uitgegaan bij het formuleren van zijn
3 axioma’s?
f) Gebruik variatie rekening om de kortste baan tussen twee punten op een oppervlak te bepalen.
g) Met welk soort van transformaties kun jij variabelen cyclisch maken?
h) Leg uit hoe jij de Feigenbaum constante δ van de chaotische slinger kunt bepalen.
Wat is het verschil voor de waarde van δ voor de chaotische slinger en de “logistic map”?
i) Wat zijn de twee postulaten van Einstein?
j) Wat zijn covariante vectoren en wat is het verschil tussen een Galilei transformatie en een Lorentz transformatie?
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(2) A classical model for small-amplitude oscillations of a linear molecule
Consider the classical mechanical model for oscillations of the linear and symmetric molecule shown in Fig. 1. Two masses m
1
= m
3
= m are connected to a third central mass m
2
= M by means of two Hookian springs with equal springconstants k . The equilibrium conformation of the ‘molecule’ is that the three masses are collinear and separated by equal distances d . We neglect the influence of gravity and external forces.
k k m M m d d
Figuur 1: Equilibrium configuration of a classical mechanical model of a linear molecule.
a) How many degrees of freedom does the system have?
b) In the absence of external forces, the oscillations of the ‘molecule’ do not depend on its position, nor on its orientation. Therefore, in order to determine the eigenfrequencies we can use a coordinatesystem in which the center-of-mass is at the origin (and remains at rest) and where the equilibrium positions of the masses lie on the x -axis. Furthermore, we can assume that the displacements of the three masses are confined to the xy -plane and that the total angular momentum J tot of the ‘molecule’ is and remains zero. How many remaining degrees of freedom and thus how many independent eigenmodes does the system have under these conditions?
c) Determine the exact kinetic and potential energies and the Lagrangian.
d) Show/argue that the approximate Lagrangian for small amplitudes is separable, leading to purely longitudinal and purely transverse eigenmodes.
e) Determine the eigenfrequencies of the longitudinal modes using det( B − ω 2
T ) = 0.
Hint: since you know that the problem is separable, assume that y
1
= y
2
= y
3
= 0, i.e. disregard the y -coordinates. Determine the approximate kinetic and potential energies T (app) , V (app) and the matrices T and B defined by the relations T (app) = t · T · q and V (app) = ~ t · B · ~ . Eliminate x
2 through the condition that the center of mass remains stationary at the origin, i.e. express x
2 as a function of x
1 and x
3
.
f) Determine the displacement vectors of the longitudinal eigenmodes and sketch them.
g) For the determination of the transverse eigenfrequency, we can eliminate y
2 again through the condition that the center of mass remains stationary at the origin.
Show that one can also eliminate y
3 as a consequence of the condition J tot
= 0.
h) In order to calculate the molecular vibration frequencies, we would have to know the (unknown) molecular force constant k . However, the ratio of the frequencies is independent of k ! For a CO
2 molecule, the mass ratio is m/M = 16 / 12 and the experimentally measured ratio of the largest and the smallest longitudinal eigenfrequencies is 1.76. What is the frequency ratio for our classical mechanical model?
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(3) Hamiltonian of a particle in a central force potential
Assume a conservative system, where a particle moves in a horizontal plane and in a central force potential U ( r ).
a) The problem is best described in terms of plane polar coordinates. Express the
Cartesian coordinates ( x, y ) and their derivatives ( ˙ y ˙ ) in terms of the plane polar coordinates ( r, θ ).
b) By determining the kinetic and potential energies, derive the Lagrangian in terms of the polar coordinates.
c) With the generalized coordinates and the Lagrangian at hand, compute the corresponding generalized momenta p r and p
θ
.
d) Derive the Hamiltonian using a Legendre transformation of the Lagrangian.
e) Show that the energy is conserved. For this, assume that in general the Hamiltonian may contain an explicit time dependence so that H = H ( q, p, t ). Write down the total time derivative of a general Hamiltonian and then rewrite it using
Hamilton’s equations in order to show that d H dt
=
∂ H
.
∂t
What does this imply? Based on this result, summarize the correct arguments allowing you to rewrite H as the sum of the kinetic and potential energy.
f) Show that the angular momentum is conserved by deriving ˙
θ from Hamilton’s equations.
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(4) The one-dimensional harmonic oscillator a) Determine the Hamiltonian of the one-dimensional harmonic oscillator.
b) Write down Hamilton’s equations for the problem and find the general solution of the problem.
(5) Special relativity
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(5 pnt) a) Σ and Σ ′ are two inertial systems. Σ ′ moves relative to Σ with a velocity v in z -direction. Two events take place in Σ at times t
1
= z
0
/c and t
2
= z
0
/ 2 c and positions ( x
1
= 0, y
1
= 0, z
1
= z
0
) and ( x
2
= 0, y
2
= y
0
, z
2
= 2 z
0
). How large does the relative velocity v have to be so that both events take place at the same time in Σ ′ ? At which time t ′ will the events be observed in Σ ′ ?
b) Σ and Σ ′ are two inertial systems moving with a relative velocity of ~v = v~e z
=const.
towards each other. A stick resting in Σ forms an angle of 45 o with the z -axis.
Under which angle does it appear in Σ ′ ?
Assume a particle in Σ has a velocity of ~u = ( v, 0 , 2 v ). Which angle does its trajectory form with the z -axes in Σ and Σ ′ ?
At last, assume a photon leaving the origin of Σ at a time t = 0 in a direction which forms an angle of 45 o with the z -axis. Under which angle does it appear in
Σ ′ ?
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