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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 Monday 21. May 2007 kl. 0900 - 1300 English version

Remedies: C

• K. Rottmann: Mathematical formulae

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

Appendix A: The questions (Page 2 - 6). Appendix B: Formulas (Page 7).

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 around June 12.

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Appendix A: The questions

QUESTION 1 [Counts 25%]

a) Derive Lagrange’s equation,

\frac{d}{dt} \frac{∂L}{∂q̇} − \frac{∂L}{∂q} = 0,

from Hamilton’s principle

δI = δ ∫_{t_{1}}^{t_{2}} L(q, q̇, t) dt = 0.

for a system with one degree of freedom. Here, there is no variation in the end points, and virtual variations in the coordinate q is performed at fixed time t, i.e., δt = 0.

b) A point mass m slides without friction on a ring which rotates counterclockwise in the xy plane with constant angular velocity ω_{0}. The ring, with radius R, rotates around a point on the ring. This point lies fixed at the origin x = y = 0, as shown in the figure below (to the left).

• Find the lagrangian L(θ, θ̇) of the mass m.

• Find Lagrange’s equation, i.e., the equation of motion for the mass m.

• The equation of motion would be the same if the mass m were swinging back and forth at the end of a massless rod in the field of gravity (see figure below, to the right; acceleration of gravity = g). Verify this, and determine the length l of the rod.

QUESTION 2 [Counts 25%]

Three spheres of mass m (sphere nr 1, to the left), 3m (sphere nr 2, in the middle), and 2m (sphere nr 3, to the right), respectively, are connected via two identical (and ideal) springs with spring constant k, as shown in the figure:

The springs can only be stretched, not bent. We assume that the spheres may only move along the x axis, and we will here investigate oscillations around the equilibrium positions of the spheres. These are x_{10}, x_{20}, and x_{30}, respectively.

• Use the deviations from equilibrium, η_{i} = x_{i} − x_{i0} (i = 1, 2, 3), as coordinates, and find the (symmetric) matrices V and T in the quadratic expressions

V = \frac{1}{2}V_{ij} η_{i} η_{j},
T = \frac{1}{2}T_{ij} η̇_{i} η̇_{j},

for the potential and the kinetic energy of the system. Here, V_{ij} and T_{ij} are matrix elements of V and T, respectively.

• Solve the secular equation

|V − ω^{2} T | = 0,

and thus determine the two eigenfrequencies of the system, f_{α} = ω_{α}/2π (α = 1, 2). Find numerical values of f_{α} when m = 100 g and k = 10^{3} N/m. (Neglect the mode with ω = 0, which corresponds to pure translation of the system.)

QUESTION 3 [Counts 25%.]

In this question, we will study a canonical transformation (q, p) → (Q, P) of a simple one-dimensional harmonic oscillator (i = \sqrt{−1}):

Q(q, p) = \frac{1}{\sqrt{2i}} (q + ip) ,
P (q, p) = \frac{−1}{\sqrt{2i}} (q − ip) .

For simplicity, we use m = k = 1 (m = mass and k = spring constant of the oscillator).

• What is the hamiltonian H(q, p) = T + V of this oscillator?

• Show that the given transformation (q, p) → (Q, P) is canonical. Hint: Use the fact that Poisson brackets are invariant under a canonical transformation, for example [q, p] = [Q, P].

• Find the hamiltonian K(Q, P) in the new coordinates Q, P. (Here, K = H.)

• Find Hamilton’s equations for Q and P, and solve these equations, using the initial conditions q(t = 0) = p(t = 0) = 1.

• What is the total energy of the oscillator?

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

a) The electromagnetic field tensor F_{μν} is defined as

F_{μν} = \frac{∂A_{ν}}{∂x_{μ}} − \frac{∂A_{μ}}{∂x_{ν}} .

Find the elements of the matrix F (i.e.: expressed in terms of the fields E and B).

b) Show that E · B is invariant under a Lorentz transformation.

c) A point charge q passes through the origin at time t = 0 and moves with constant velocity v in the positive x direction. Let us by S_{0} denote the inertial system where the point charge is at rest, and by S we denote the inertial system where the point charge moves with velocity vx̂ (x̂ is a unit vector). The electric field at a distance r from the point charge is

E = \frac{q}{4πε_{0}} \frac{1 − β^{2}}{(1 − β^{2} sin^{2} θ)^{3/2}} \frac{r}{r^{3}}

measured in S. Here, θ denotes the angle between the x axis and r, see figure below, and β = v/c. What is the magnetic field B_{0} measured in S_{0}? Show that the magnetic field, measured in S, may be written as

B = \frac{1}{c^{2}} v × E.

d) What is the magnetic field B in the nonrelativistic limit v ≪ c? Compare the result with Biot-Savart’s law,

dB = \frac{μ_{0}}{4π} \frac{I dl × r}{r^{3}} .

Sketch field lines for B in the yz plane.

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

a) Let us investigate how the stock of a certain animal will develop in time. Assume - in the simplest possible manner - that this number increases with a constant factor each year, i.e., there exists a μ > 1 such that

b_{n+1} = μ b_{n},

where b_{n} is the stock in year n and b_{n+1} is the stock in year n + 1. What kind of equation is this, and what is its general solution? Explain briefly why this model is unrealistic.

b) An improved model is obtained by letting B be an upper limit on the number of animals. Define the model by

b_{n+1} = μ b_{n} (1 − \frac{b_{n}}{B} ) .

We introduce the relative stock,

x_{n} = \frac{b_{n}}{B} ,

and hence we have

x_{n+1} = μ x_{n} (1 − x_{n}) .

Next, we define

F_{μ}(x) = μ x (1 − x) ,

which yields

x_{n+1} = F_{μ}(x_{n}).

Make a simple sketch of F_{μ}(x), and write down the coordinates of the maximum point of the curve. Let x_{0} be the intersection between the graph y = F_{μ}(x) and the graph y = x. Then, what is x_{n} as a function of x_{0}? What kind of point is x_{0}?

c) Define the concept ”fixed point” for a function f. Define the concept ”periodic point” for a function f, and explain what is meant by ”fundamental period” for the point. Explain/define ”attractive fixed point” and ”repulsive fixed point” for a function f.

d) Finally, consider the function

F_{μ}(x) = μ x (1 − x) .

Determine the two fixed points of F_{μ}. Let the upper limit of μ be 3, and determine whether the fixed points are attractive or repulsive.

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Appendix B: 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

• Poisson brackets:

[f, g] = \frac{∂f}{∂q_{i}} \frac{∂g}{∂p_{i}} − \frac{∂g}{∂q_{i}} \frac{∂f}{∂p_{i}}

[q_{i}, q_{j}] = [p_{i}, p_{j}] = 0 , [q_{i}, p_{j}] = δ_{ij}

• Four-vector:

x_{μ} = (r, ict)

• 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

β = v/c , γ = 1/\sqrt{1 − v^{2}/c^{2}}

• Lorentz transformation of electromagnetic field (the inertial system S_{0} moves with velocity vx̂

relative to S):

E_{x} = E_{0x} , E_{y} = γ(E_{0y} + vB_{0z}) , E_{z} = γ(E_{0z} − vB_{0y})
B_{x} = B_{0x} , B_{y} = γ(B_{0y} − \frac{v}{c^{2}} E_{0z}) , B_{z} = γ(B_{0z} + \frac{v}{c^{2}} E_{0y})

• Trigonometric relations:

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