Karlstads Universitet Fysik ANALYTISK MEKANIK HT 2011 Kursens kod: FYGC04 Undervisande lärare: Jürgen Fuchs rum 21F 316 tel. 054-700 1817 el.mail: jfuchs@fuchs.tekn.kau.se Examination: Inlämningsuppgifter ( 30 p.) + studentpresentationer ( 20 p.) För betyget godkänt på kursen krävs godkända inlämningsuppgifter (minst 15 poäng) samt godkänt presentation (minst 10 poäng). För betyget väl godkänt krävs 37.5 poäng, varav minst 10 poäng för presentation. Syfte och mål: Att tillägna fördjupade kunskaper såväl konceptuella som tillämpbara i klassisk mekanik, speciellt inriktade mot Lagrange- och Hamiltonformalismerna, vilka även har tillämpningar utanför den klassiska mekaniken. Dessutom att förberada för forskarstudier. Innehåll: Mekanikens matematiska struktur: Lagranges och Hamiltons ekvationer, variationsprinciper, symmetrier och konserveringslagar, kanoniska transformationer, Hamilton--Jacobi teori och lösbarhet. Tillämpningar: Tvåkroppsproblemet, stelkroppskinematik, stelkroppsdynamik, små svängningar. Kurslitteratur: H. Goldstein, C.P. Poole and J.L. Safko, Classical Mechanics (3rd edition), Benjamin-Cummings 2002 [ISBN 0 201 65702 3] Kursens omfattning: Goldstein- Poole- Safko utan Kapitel 7.5-7.11, 8.4, 11.5-11.9, 12, 13 FYGC04 HT 2011 Exercises 1 2012-01-09
Analytisk Mekanik(FYGC04) HT 2011 1. [ utdelning: 11-11-11 inlämning: 11-11-21 ] 5p. a) Show that the Euler--Lagrange equations do not change when a total time derivative d F(q,t) is added to the Lagrangian. dt b) A point mass of mass m is connected by an inextensible massless wire with a spring of spring constant k, which in turn is attached to a vertical wall. Between the point mass and the spring, the wire partly runs along a cylindrical wheel, in such a way that the point mass is hanging vertically, while the spring is horizontal. When the point mass moves vertically, the wheel rotates correspondingly around its cylinder axis. Determine the extension l of the spring at equilibrium, compared to its extension l in the absence of the mass. Find the motion of the system, assuming that at time t = 0 the extension of the spring is l(t=0) = l. Hint: 1) A convenient generalized coordinate is d := l l. 2) The system is conservative. ( 1 p.) c) A double pendulum consists of an ordinary pendulum of length l 1 and mass m 1 attached to a fixed point in space and a second pendulum of length l 2 and mass m 2 attached to the end point of the first pendulum. Explain why, with the usual idealizations (i.e., each pendulum consisting of an inextensible rod with a point mass at the end, no friction, etc.) the number of degrees of freedom is 2 when the motion is confined to a vertical plane. How many degrees of freedom are there if the motion is not confined to a plane? Assuming that the motion is in a plane, find the equations of motion in terms of appropriate angles θ 1 and θ 2 (e.g. those used in figure 1.4 on p. 14 of [Goldstein--Poole--Safko]. FYGC04 HT 2011 Exercises 2 2012-01-09
d) Fermat s principle in classical optics states that light always travels along the path that takes the least time. Consider a light beam travelling within a vertical plane, in a medium with index of refraction n depending on the vertical position z, n = n(z). Use the calculus of variations to minimize the total time needed for the beam to travel between two given points, and thereby show that the function z=z(x), with x the coordinate in horizontal direction, satisfies the differential equation d 2 z dx = 1 dn [ ( dz ) 2 ] 1+. 2 n(z) dz dx e) Determine the equation of motion for a point particle on which both the gravitational force and a friction force of magnitude F fr = k v act. Show that when the particle starts at rest, its speed v cannot exceed mg/k. FYGC04 HT 2011 Exercises 3 2012-01-09
Analytisk Mekanik(FYGC04) HT 2011 2. [ utdelning: 11-11-15 inlämning: 11-11-28 ] 5p. a) Derive the Euler equation for the variational problem F(y,y,y,x)dx) = 0 δ( (with prescribed values of both y and y dy dx at the end points x 0 and x 1 ). b) Considerthemotionofamasspointundertheinfluenceofgravityonafrictionless inclined plane. Instead of(being smart and) using a single generalized coordinate, describe the problem in terms of the two Cartesian coordinates x (horizontal) and z (vertical) and account for the constraint by a Lagrange multiplier. Set up the equations of motion, solve them, and identify the constraint force. c) Two point masses m 1 and m 2 areconnected by a (massless, inextensible) ropeof length l which passes through a small hole in a horizontal plane. The first point mass moves (without friction) on the plane, while the second point mass oscillates like a simple planar pendulum in a constant gravitational field of strength g. Using two angles and the lengths of the two parts of the rope as variables, write down the Lagrangian of the system. Find a constraint and thereby determine the number of degrees of freedom. What is the physical significance of the constraint force? Eliminate the constraint (either directly, or via a Lagrange multiplier) and obtain the equations of motion. Try to find some special solution(s) to the equations of motion. Find two conserved quantities. (2p.) d) A particle is moving in a central potential V = V(r) in such a way that its orbit is a circle that passes through the origin r=0. Describe the orbit as a suitable function r = r(θ). Starting from this function, reconstruct the potential. ( 1 p.) FYGC04 HT 2011 Exercises 4 2012-01-09
Analytisk Mekanik(FYGC04) HT 2011 3. [ utdelning: 11-11-25 inlämning: 11-12-05 ] 6p. a) Consider the one-dimensional motion of a particle in the potential V = V(x) = D(e 2ax 2e ax ), where D and a are positive constants and x R. (This is a one-dimensional version of the so-called Morse potential, a central potential that phenomenologically describes forces between the atoms in diatomic molecules.) Show that the motion is bounded iff E is negative, find the minimum possible value of E, and determine the turning points. Obtain expressions for x as a function of time, separately for the cases that the total energy E is positive, zero, or negative. b) A point mass is moving in a central force field. Its orbit has the form r(θ) = ae cθ with positive constants a and c. Draw this orbit (qualitatively). Determine the force that allows such an orbit and obtain the corresponding potential V(r), as well as the effective potential V eff (r). Discuss qualitatively the motion for different values of the energy, using a plot of V(r) and V eff (r). For the orbit given above, determine also the motion in time, i.e. the functions r(t) and θ(t). c) Compute the differential scattering cross section σ(ϑ) dϑ for scattering in the central potential V = V(r) = k r 2. FYGC04 HT 2011 Exercises 5 2012-01-09
d) Compute the moment of inertia tensor I with respect to the origin for a cuboid of constant mass density whose edges (of lengths a,b,c) are along the x,y,zaxes, with one corner at the origin. For a=b=c, i.e. a cube, find the principal axes and the principal moments of inertia. Determine the angular momentum L and the angle between L and ω when thecube is rotatingwith angular velocity ω about oneof the coordinateaxes. e) A rigid body of the form of a homogeneous straight stick (of mass M, length l and infinitesimal thickness) is placed in a homogeneous gravitational field. The stick is held fixed at one of its ends, but otherwise allowed to move freely. Give the Lagrangian of the system in spherical coordinates for a coordinate system with origin at the fixed point. Obtain the Euler--Lagrange equations. Solve them partially in the form of an orbit equation ϕ = ϕ(θ) for the particular initial condition that at time t=0 the stick is rotating in the plane perpendicular to the gravitational force. Determine the turning points for the motion obtained in the previous point. f) A homogenous cone of total mass M, height h and base radius R is rolling without slipping on a horizontal plane, with angular velocity of constant magnitude ω. Compute the kinetic energy. Hints: Since the line along which the cone touches the plane is temporarily at rest, the direction of the angular velocity vector is given by this line. The center of mass of the cone is located along its symmetry axis at distance 3h/4 from the top. In the body system, with origin at the center of mass and x -direction along the symmetry axis, the moment of inertia tensor is diagonal, with diagonal 3 entries 10 MR2 3, 80 M (4R2 +h 2 3 ), 80 M (4R2 +h 2 ). FYGC04 HT 2011 Exercises 6 2012-01-09
Analytisk Mekanik(FYGC04) HT 2011 4. [ utdelning: 11-12-04 inlämning: 12-01-06 ] 7p. a) A bead slides without friction in a uniform gravitational field on a vertical circular hoop. The hoop is rotating at constant angular velocity about its vertical diameter. Denote the mass of the bead by m, the radius of the hoop by R, the angular velocity of the hoop by ω, and the angle between the vertical and the line that connects the bead with the origin of the hoop by ϑ (in such a way that ϑ=0 when the bead is at the bottom of the hoop). Write down the Lagrangian L as a function of ϑ and ϑ. Determine the equilibrium values of ϑ as a function of ω and discuss their stability. Find the frequencies of small vibrations about the stable equilibrium positions. Discuss in particular the case that the rotation frequency has the special value ω= g/r, with g the gravitational acceleration. (2p.) b) Consider a pendulum consisting of two parts: a uniform rod of mass m, length l, negligible thickness, and with one end fixed; and a uniform disk of mass µ and radius ρ. Therodismovinginaplane, andthediskisattachedatapointp onitsboundary to the non-fixed end of the rod, in such a way that it can freely rotate about P in the plane in which the rod is moving. Obtain the Lagrangian and the equations of motion. Give the kinetic and potential energy matrices in the limit of small deviations from stable equilibrium. ( 1 p.) c) A point mass m experiences a force F( r) = k r e 2 r mg e z (r= x 2 +y 2 +z 2 ). Determine the potential and find a physical interpretation for it. Introduce coordinates (u, v, ϑ) that are determined by x = uv cosϑ, y = uv sinϑ, z = 1 2 (u v). Obtain the Lagrangian and the Hamiltonian in these coordinates, and find at least two constants of motion. FYGC04 HT 2011 Exercises 7 2012-01-09
d) Recall from electrodynamics that the force on a point particle of charge q and velocity v exerted by an electric field E and a magnetic field B is the Lorentz force F = q( E + v c B). Show that the equation of motion for the particle can be deduced from the Lagrangian (1.63), i.e. from L = 1 2 mv2 qφ+ q c v A. Hint: 1) Do not confuse d A dt with A t. 2) To arrive at the term v ( A) you can either combine the two terms in the bac-cab rule for double cross products, or work with the Levi-Civita symbol ǫ αβγ. e) Starting from the Lagrangian (1.63), obtain the Hamiltonian for a charged particle in an electromagnetic field in terms of the potentials φ and A. Deduce the Hamilton equations of motion for the particle and show that, when written in terms of the electric field E and magnetic field B, they give the formula for the Lorentz force. f) Replace the Lagrangian (1.63) by L = mc 2 1 v2 c qφ+ q 2 c v A. In which sense does this generalize L? Determine the generalized momenta, the Hamiltonian H, and the Hamilton equations of motion for H. Is H conserved? Can it be written in the form T +V? FYGC04 HT 2011 Exercises 8 2012-01-09
Analytisk Mekanik(FYGC04) HT 2011 5. [ utdelning: 11-12-07 inlämning: 12-01-09 ] 7p. a) Using spherical coordinates (r, ϑ, ϕ), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p ϕ (0)=0 and ϕ(0)=0. b) Consider the system ẍ j +ω 2 j x j = 0 for j=1,2,...,n of differential equations, with parameters satisfying ω j ω k for j k. By regarding the coordinates x j as the components of an n-component vector x, this describes an anisotropic harmonic oscillator in the n-dimensional space R n. Find a force F = F( x, x) parallel to x that constrains x to stay on the unit sphere S n 1 ={ y R n y =1} in R n. Write down the equations of motion for the resulting constrained oscillator. Show that the n functions (ẋ j x k x j ẋ k ) 2 f j := x 2 j + ω 2 k j j ω2 k are conserved quantities. Hint: Show that x S n 1 implies that j x jẍ j is proportional to jẋ2 j, and use this result to derive from the equations of motion a formula for F / x, which then in turn can be used to rewrite the equations of motion in a nicer form. c) In view of the different signs appearing in the two subsets of the Hamiltonian equations of motion, one may try to combine them by using complex quantities. Show that, for a system with one degree of freedom, the transformation (q,p) (Q,P) with Q := q +ip and P := q ip (with unaltered Hamiltonian) is not canonical. Can the transformation be made canonical by multiplying P and/or Q by numerical factors which are real or purely imaginary numbers? FYGC04 HT 2011 Exercises 9 2012-01-09
d) Write down the Hamilton-Jacobi equation for a two-dimensional central force in polar coordinates. Show that in these coordinates the equation is separable and obtain Hamilton s principal function. Deduce from the result that angular momentum is conserved. e) Consider one-dimensional motion of a point particle under the influence of an external force F =F(t) which isanexplicit functionoftime, but doesnot depend on the position x or velocity ẋ of the particle. Set up the Hamilton-Jacobi equation, for F an arbitrary function of t. Solve the Hamilton-Jacobi equation for S(x, P, t) in the specific case F(t) = c sin(ωt) by making the separation ansatz S(x,P,t) = S 0 (t)+xs 1 (t). Taking the so obtained function S as the generating function for a canonical transformation, obtain the motion of the particle subject to the initial conditions x(t=0) = 0 = ẋ(t=0). f) FYGC04 HT 2011 Exercises 10 2012-01-09
Consider small vibrations of a system about steady motion, using the Hamiltonian formalism for its description. Show that generalized coordinates can be chosen in such a way that at the point of steady motion both the coordinates and the momenta are constant. Show that to lowest non-vashing order in the deviation from steady motion the Hamiltonian H can be written as H = 1 2 ξkξ with ξ=η η 0 and K=K(η 0 ) a matrix whose entries only depend on η 0, where η (q 1,q 2,...,q N, p 1,p 2,...,p N ) t and η 0 corresponds to the steady motion. Denote by M a matrix whose columns are eigenvectors of K, corresponding to the eigenfrequencies ±ω j of small vibrations. For the case that all frequencies ω j are distinct, show that the normalizations of the eigenvectors can be chosen in such a way that M has the properties of the Jacobian matrix for a canonical transformation, and that after this canonical transformation the new Hamiltonian H (q,p ) takes the form H = i N j=1 ω jq jp j. Write down the equations of motion for this new Hamiltonian. Finally show that a further canonical transformation corresponding to the choice N N F 2 = q jp j + i ( 2 ωj 1 Pj 2 1 ) 2 ω jq j 2 j=1 j=1 of generating function decomposes H into the sum of N Hamiltonians describing uncoupled harmonic oscillators. ( 2 p.) FYGC04 HT 2011 Exercises 11 2012-01-09