ANALYTISK MEKANIK HT 2013

Size: px

Start display at page:

Download "ANALYTISK MEKANIK HT 2013"

George Douglas
5 years ago
Views:

Karlstads Universitet Fysik ANALYTISK MEKANIK HT 2013 Kursens kod: FYGC04 Undervisande lärare: Jürgen Fuchs rum 21F 316 tel. 054-700 1817 el.mail: jfuchs@fuchs.tekn.kau.

1 Karlstads Universitet Fysik ANALYTISK MEKANIK HT 2013 Kursens kod: FYGC04 Undervisande lärare: Jürgen Fuchs rum 21F 316 tel el.mail: 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 ] Kursens omfattning: Goldstein- Poole- Safko utan Kapitel , 8.4, , 12, 13 FYGC04 HT 2013 Exercises

2 Analytisk Mekanik(FYGC04) HT [ utdelning: inlämning: ] 5p. a) Consider a system of N point masses m i (i=1,2,...,n, with N arbitrary), with a total force F i tot acting on the ith mass. Show that the center of mass R c.m. :=( N i=1 m i r i )/( N moves like a single particle of mass i=1 m i) N m i acted upon by a force F c.m. = i=1 N i=1 F tot i. b) Show that the Euler--Lagrange equations do not change when a total time derivative d F(q,t) is added to the Lagrangian. dt c) The pressure p(r) and density ρ(r) of a self-gravitating fluid that is distributed in a spherically symmetric way obey the differential equation ( ) d r 2 dp = 4πGr 2 ρ, where G is the gravitational constant. dr ρ dr Solve this equation subject to the boundary condition that p=ρ=0 for all values of r larger than some value R, separately for the following two cases: ρ = ρ 0 = const (for r R) and p = cρ 2 with c=const. To what physical systems might these solutions be approximations? d) Consider a system described by the most general form for a purely kinetic Lagrangian, i.e. by L = 1 2 N i,j=1 g ij (q) q i q j with g an arbitrary (suitably well behaved) function of the generalized coordinates q k. Determine the Euler--Lagrange equations of the system. Find situations in which differential equations of the form obtained this way are important. Hint: As a relevant keyword you can try general relativity. FYGC04 HT 2013 Exercises

3 e) Consider the function f(y,ẏ;x) = 1 2 M(y)ẏ2 V(y), where V is an arbitrary function of y, M is a positive function of y, y is a function of x, and ẏ= dy dx. Show that the integral J[y] := x2 x 1 f(y,ẏ;x)dx, with prescribed boundary values y(x 1 )=y 1 and y(x 2 )=y 2, attains a minimum (i.e., not just an extremal value) for y a solution to the stationarity condition f y d ( f ) = 0. dx ẏ Hint: Writing, as usual, y(x)=y (x)+αη(x), show that the second derivative d2 J(α) dα 2 positive at points at which dj dα vanishes. Compute d2 J dα 2 without performing a partial integration in dj dα. is FYGC04 HT 2013 Exercises

4 Analytisk Mekanik(FYGC04) HT [ utdelning: inlämning: ] 7p. a) Consider a bar-bell two point masses m 1 and m 2 connected by a massless inextensible rod. The point mass m 1 is constrained to move along a horizontal line, while the point mass m 2 can move in the vertical plane that contains this line. Apart from the constant vertical gravitational force acting on the masses there are no other applied forces. Write down the constraints and determine the number of degrees of freedom. Introduce suitable generalized coordinates and determine the Euler--Lagrange equations of the system. Simplify the equations of motion as much as possible by invoking conservation laws and the initial conditions that at t=0 the mass m 1 is at the origin, m 2 is vertically beow it, and the angular velocity of m 2 with respect to the origin is ω. Thereby obtain an orbit equation for the motion of m 2. Describe the geometry of the resulting motion of the point massm 2. Solve the equations of motion approximately for the situation that the orientation of the bar-bell is close to vertical. b) If the integrand f in the variational problem δ( f dx)=0 (the analogue of the Lagrangian) does not depend explicitly on the independent variable x (the analogue of the time t), then the Euler equation can be integrated to ẏ f ẏ f = const. Use this information to prove that the line of shortest distance between two points on a sphere forms a part of a great circle. Hint: Use spherical coordinates and let the angle ϕ play the role of x. c) 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. FYGC04 HT 2013 Exercises

5 d) A ladder is leaning on a horizontal floor against a vertical wall. Due to its weight, the ladder will slide down the wall and along the floor. Aftersometime t c theladderwilllosecontactwiththewall. Beforethishappens, the system has one degree of freedom, say the angle θ between the ladder and the floor. Describe the motion for t t c, assuming that the wall and floor are frictionless: Express the kinetic energy T through θ and θ. Introduce a constraint force N and write down the equations of motion for the center of mass position and for the angle θ. Then eliminate (the components of) the constraint force from the equation of motion for θ to obtain an uncoupled differential equation for θ. Use energy conservation to determine the angle θ c =θ(t c ) and thereby obtain an expression for the time t c. Hint: To determine T, either regard the ladder as a rigid body, or proceed as follows: divide the ladder into infinitesimal pieces of length dl and of mass dm=(m/l)dl (with L the length and M the mass of the ladder), and obtain T by summing up the contributions from all these pieces. ( 2 p.) e) A uniform circular inextensible hoop can roll frictionless and without slipping on a horizontal plane along a fixed direction. A point particle is constrained to move along the inside of the hoop. The particle and the hoop are acted upon by a constant vertical gravitational force. Consider the situation that at some initial time t=0 the hoop is at rest and the particle is located at the top of the hoop and has horizontal velocity v top. Using the Lagrangian formalism, compute the horizontal velocity v bot of the particle at the time when it first passes the bottom of the hoop. Consider the result specifically in the limits that the mass of the particle is either much smaller or much larger than the one of the hoop. f) Show that every orbit in the central potential V( r) = k 2 r 2 with k>0 (the three-dimensional harmonic oscillator) is an ellipse centered at r = 0. ( 1 p.) FYGC04 HT 2013 Exercises

6 Analytisk Mekanik(FYGC04) HT [ utdelning: inlämning: / ] 7p. a) The orbit of the comet Hyakutake around the sun has an eccentricity e = , and its perihelion is.2302 AU. The last time the comet appeared on the sky was the period from March to May When will it appear next time? Compute the aphelion and compare it to the one of the (former) planet Pluto. (Some astronomical data: Astronomical unit: 1AU= meter; solar mass: M = kg; semimajor axis of Pluto s orbit: 39.37AU; eccentricity of Pluto s orbit:.249. The comet s mass is negligible compared to M.) b) 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). With the help of the orbit equation, reconstruct the potential from this function r(θ). c) Compute the differential scattering cross section σ(ϑ) dϑ for scattering in the central potential V = V(r) = k r 2. d) Consider a homogeneous ball constrained to move on a horizontal plane. Show that when the ball is rolling without slipping on the plane, then the total horizontal force on the ball must be zero. Hint: Consider the ball as a rigid body and suitably combine the Newtonian equations of motion describing the translation of the center of mass and the rotation about the center of mass with the (nonholonomic) rolling constraints which connect the translation and rotation. FYGC04 HT 2013 Exercises

7 e) A homogeneous 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 ). f) A rigid body of the form of a homogeneous straight stick of infinitesimal thickness is placed in a homogeneous gravitational field. One end of the stick is constrained to move (without friction) along a horizontal line and is connected to a horizontal spring which exerts a force along that line. The stick can swing in the vertical plane that contains the horizontal line. Give the Lagrangian of the system and obtain the Euler--Lagrange equations. In the approximation of small vibrations about the stable equilibrium of the system, compute the normal frequencies. ( 1 p.) g) Considertwoevents whichinaninertialsystem S (havingonetimeandonespace dimension), happen at the same time t 1 =t 2 =l/2c and at positions x 1 = l/2 and x 2 =+l/2. respectively. Compute the time difference t =t 2 t 1 between the two events that is measured in another inertial system S that moves at a velocity v in x-direction with respect to S. (The result us non-zero, showing that the two events, which are simultaneous in system S, are not simultaneous in system S.) Compute the spatial distance x =x 2 x 1 between the two events that is measured in S. Does the distance x describe the length of any physical object? FYGC04 HT 2013 Exercises

8 Analytisk Mekanik(FYGC04) HT [ utdelning: inlämning: ] 5p. a) 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. b) 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? c) 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. FYGC04 HT 2013 Exercises

9 d) Consider the time-dependent Hamiltonian H(q,p;t) = with constant m and ω. p 2 2m sin 2 (ωt) ωpq cot(ωt) m 2 ω2 sin 2 (ωt)q 2 Find a corresponding Lagrangian L=L(q, q;t). Then obtain, by choosing a suitable new coordinate q, an equivalent time-independent Lagrangian L. Determine the new Hamiltonian H that corresponds to this new Lagrangian. Discuss similarities and differences between H and H. e) Usingsphericalcoordinates (r,ϑ,ϕ), obtainthehamiltonianandthehamilton 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. FYGC04 HT 2013 Exercises

10 Analytisk Mekanik(FYGC04) HT [ utdelning: inlämning: ] p. a) Study the motion of a point particle in a central potential of the form V(r) = a r + b r 2 Show that the orbits have the form d r(θ) = 1+ε cos(αθ) with suitable constants d, ε and α. with a>0. What do the orbits look like when α=1+η with η 1? (Discuss separately the cases when ε>1, ε=1 and ε<1.) b) A point mass m experiences a force F( r) = k r 2 e 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. c) The suspension point of a pendulum (wich, as usual, is assuemd to consist of a mass point at the end of a massless inextensible rod) is placed on a horizontal wire, on which it can slide without friction. Determine the Lagrangian as a function of the position of the suspension point along the wire and of the angle of the pendulum with respect to the vertical. Obtain the conjugate momenta to the latter variables. Obtain the Hamiltonian by the usual Legendre transformation and write down the Hamilton equations of motion. Give all cyclic coordinates and the corresponding conserved quantities. ( 1 p.) FYGC04 HT 2013 Exercises

11 d) Consider a particle moving on a sphere (say, on the surface of the Earth) under the influence of a conservative force with potential V. Write down the Hamiltonian using spherical coordinates ϑ, ϕ on the sphere. Assume that the sphere is rotating about the z-axis with angular velocity ω. Then besides the previously used coordinates ϑ, ϕ which describe the particle in a non-rotating coordinate system, also a rotating coordinate system ϑ,ϕ is of interest, where ϑ = ϑ and ϕ = ϕ+ωt. Determine the generating function that affords the canonical transformation (ϑ, p ϑ, ϕ, p ϕ ) (ϑ, P ϑ, ϕ, P ϕ ) and obtain the Hamiltonian in rotating coordinates. ( 1 p.) e) 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? f) Consider a canonical transformation with generating function F 2 (q,p) = qp +εg 2 (q,p), where ε is a small parameter. Write down the explicit form of the transformation. Neglecting terms of order ε 2 and higher, find a relation between this transformation and Hamilton s equations of motion, by setting G 2 =H (why is this allowed?) and ε=dt. g) 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. Read off from the result that angular momentum is conserved. FYGC04 HT 2013 Exercises

ANALYTISK MEKANIK HT 2011

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