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NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET INSTITUTT FOR ENERGI- OG PROSESSTEKNIKK

Contact during exam: Jon Andreas Støvneng Phone: 73 59 36 63 / 45 45 55 33

EXAM TEP4145 KLASSISK MEKANIKK Thursday 16. August 2007 kl. 1200 - 1600 English version

Remedies: C

• K. Rottmann: Mathematical formulae

• Approved calculator, with empty memory, according to list worked out by NTNU. (HP30S or similar.)

Page 2 - 6: The questions Page 7: Some formulas

The exam consists of 5 questions. The weight for each question is given. Do questions 1, 2 and 3, and either question 4 (physics) or 5 (cybernetics). Note that Einstein’s summation convention is used unless otherwise stated, i.e., summation over repeated indices.

The grades will be ready in August.

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QUESTION 1 [Counts 30%]

A spherical pendulum consists of a mass m attached to the end of a massless rod of length l. The other end of the rod is attached at the origin. In other words, the mass may move on a spherical shell of radius l, hence the name. The system is in the gravitational field.

a) Show that the Lagrangian of the system is

L = \frac{ml^{2}}{2} (θ̇^{2} + ϕ̇^{2} sin^{2} θ) + mgl cos θ

when we choose V = 0 in z = 0, and use generalized coordinates θ and ϕ as shown in the figure.

b) Explain how you immediately can say that the canonical momentum p_{ϕ} is a constant of the motion. Show that p_{ϕ} is identical to the z-component of the angular momentum, i.e., (r × mv)_{z}.

c) Introduce the constant

γ = ϕ̇ sin^{2} θ = \frac{p_{ϕ}}{ml^{2}}

and show that Lagrange’s equation for the coordinate θ becomes

θ̈ + \frac{g}{l} sin θ − γ^{2} \frac{cos θ}{sin^{3} θ} = 0

d) One possibility is that the mass moves in a horizontal circle, at constant θ = θ_{0}. Show that the period (time of revolution) then becomes

T_{ϕ} = 2π\sqrt{\frac{l cos θ_{0}}{g}}

e) Another possibility is that the mass ”oscillates” around θ_{0}, i.e., θ(t) = θ_{0} + α(t). Assume small oscillations, i.e. |α| ≪ θ_{0} (and hence |α| ≪ 1), and show that the motion of the pendulum is described by the equation

α̈ + α \frac{g}{l} cos θ_{0} (4 + tan^{2} θ_{0}) = 0

f) What is the oscillation period T_{α}? Calculate the ratio between the time of revolution and the oscillation period, i.e., T_{ϕ}/T_{α}, and sketch this ratio as a function of the angle θ_{0}.

QUESTION 2 [Counts 30%]

A particle of mass m moves in a spherically symmetric potential V (r). It can be shown that the motion is in a plane perpendicular to the direction of the angular momentum L, which is a constant of the motion, with |L| = |r × p| = l.

a) Determine the Lagrangian L = T − V of the particle, using the polar coordinates r and θ as generalized coordinates. (Don’t confuse the Lagrangian L with the angular momentum L.)

b) Show that Lagrange’s equation for θ becomes

\frac{d}{dt} (mr^{2}θ̇) = 0.

Show that this expresses conservation of the angular momentum.

c) Show that Lagrange’s equation for r becomes

mr̈ − mrθ̇^{2} = f(r),

where f(r) = −∂V/∂r is the force acting on the particle.

d) Use the equations and the information given above to derive the following differential equation for u = 1/r:

\frac{d^{2}u}{dθ^{2}} + u = −\frac{m}{l^{2}u^{2}} f(1/u)

e) If the particle moves in a 1/r-potential, the path will be described by the conic section

r(θ) = \frac{α}{1 + ε cos θ},

where α and ε are constants. Use the differential equation of the orbit in d) to verify this. (Assume V (∞) = 0.)

f) Next, assume that the particle follows the spiral orbit

r(θ) = r_{0}e^{θ},

where r_{0} is a constant. Find the potential V (r) in which this particle moves. (Also here, assume V (∞) = 0.)

Page 4 of 7 QUESTION 3 [Counts 15%.] A one-dimensional anharmonic oscillator may be described with the Lagrangian

L(x, ẋ) = \frac{1}{2}ẋ^{2} − \frac{1}{2}ω^{2}x^{2} − αx^{3} + βxẋ^{2}

where the mass of the oscillator equals 1.

a) What is the Hamiltonian H(x, p) of the anharmonic oscillator? A one-dimensional harmonic oscillator may be described with the Lagrangian

L_{0} = \frac{1}{2}ẋ^{2} − \frac{1}{2}ω^{2}x^{2}

and the Hamiltonian

H_{0} = \frac{1}{2}p^{2} + \frac{1}{2}ω^{2}x^{2}.

b) Write

L = L_{0} + δL

and

H = H_{0} + δH

for the anharmonic oscillator, and show that for small oscillations (i.e., |αx| ≪ ω^{2} and |βx| ≪ 1), the ”extra terms” in L and H are equal, but with opposite signs, i.e.,

δL = −δH.

QUESTION 4 [For physics students, counts 25%.]

A possible Lagrangian for a relativistic particle with mass m, moving in a conservative potential V (x_{i}), is

L = −\frac{mc^{2}}{γ} − V

a) Show that this Lagrangian results in Lagrange equations corresponding to a generalized version of Newton’s 2. law,

\frac{d}{dt} p_{i} = F_{i}

Here, p_{i} are spatial components (i = 1, 2, 3) of the four-momentum p_{μ}.

b) Show that the Hamiltonian

H = v_{i}p_{i} − L

then equals the total energy of the particle.

Let the particle be an electron with mass m = m_{e} and charge q = −e, starting at the origin with zero velocity at time t = 0, and let the potential represent a uniform electric field pointing along −x̂, i.e.,

V (x) = qE_{0}x

c) Write down the Lagrange equation for the electron and show that its speed becomes

ẋ = \frac{eE_{0}t/m_{e}}{\sqrt{1 + (eE_{0}t/m_{e}c)^{2}}}

d) Solve this equation and find the path x(t) of the electron. Check that your answer makes sense, both for small and for large values of t.

QUESTION 5 [For cybernetics students, counts 25%.]

In Norwegian only.

a) Den aller enkleste differensligningen er gitt ved relasjonen

a_{n+1} = r a_{n},

der n ≥ 0 mens r er et reelt tall. Anta at verdien a_{0} er kjent, og skriv ned uttrykket for den generelle a_{n}.

b) En representasjon av den lineære, homogene differensligningen av annen orden er

a_{n+1} = b a_{n} + c a_{n−1},

der n ≥ 1 mens b og c er reelle tall. Sett som prøveløsning

a_{n} = C r^{n},

og utled direkte fra differensligningen den karakteristiske annengradsligningen.

c) Anta at det er to distinkte røtter r_{1} og r_{2} for annengradsligningen i punkt b). Det oppgis at alle følger

a_{n} = c_{1} r_{1}^{n} + c_{2} r_{2}^{n},

der c_{1} og c_{2} er konstanter, er løsninger av

a_{n+1} = b a_{n} + c a_{n−1}.

Sett opp et system for de to ukjente koeffisientene c_{1} og c_{2}, og bestem eksplisitte uttrykk for c_{1} og c_{2}, dvs uttrykt ved de kjente størrelsene r_{1}, r_{2}, a_{0} og a_{1}. (HINT: Bruk gjerne Cramers regel ved løsning av systemet med de to ukjente.)

d) Betrakt følgen {F_{n}} av Fibonaccitall, definert ved

F_{n+1} = F_{n} + F_{n−1},

der F_{0} = F_{1} = 1. Gitt at

F_{n} = C_{1} r_{1}^{n} + C_{2} r_{2}^{n},

bestem de to distinkte røttene r_{1} og r_{2} ved å løse den tilhørende karakteristiske annengradsligningen. (HINT: Denne ligningen kan løses direkte ut fra analysen i punkt b).)

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Some formulas

The validity of the formulas and the meaning of the symbols are assumed to be known.

• Hamilton’s equations:

q̇_{i} = \frac{∂H}{∂p_{i}} , ṗ_{i} = −\frac{∂H}{∂q_{i}}

• Lagrange’s equations:

\frac{d}{dt} \frac{∂L}{∂q̇_{i}} − \frac{∂L}{∂q_{i}} = 0

• The Lorentz factor:

γ = 1/\sqrt{1 − β^{2}} , β = v/c

• Four-vector:

x_{μ} = (r, ict)

• Proper time τ defined by:

dx_{μ} dx_{μ} = −c^{2} dτ^{2}

• Four-velocity:

u_{μ} = \frac{dx_{μ}}{dτ} = γ(v, ic)

• Four-momentum:

p_{μ} = m u_{μ} = γ(mv, imc)

• Four-potential:

A_{μ} = (A, iϕ/c)

• Electromagnetic field:

E = −∇ϕ − \frac{∂A}{∂t}

B = ∇ × A

• Lorentz transformation (with relative velocity v = vx̂): L_{22} = L_{33} = 1 , L_{11} = L_{44} = γ , L_{14} = −L_{41} = iβγ

• Trigonometric relations:

cos(a ± b) = cos a cos b ∓ sin a sin b
sin(a ± b) = sin a cos b ± cos a sin b

• For |x| ≪ 1 we have:

(1 + x)^{n} ≃ 1 + nx cos x ≃ 1 sin x ≃ x