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### probset1

Course: PHYS 612, Fall 2008
School: Oregon
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Word Count: 429

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612 THEORETICAL PH MECHANICS Problem Assignment # 1 Winter 2009 01/07/09 due 01/14/09 1. Pendulum (Part 1) note: This problem will be continued in later assignments. Consider a mathematical pendulum (mass m, length ) with Lagrangian L= m 2 2 2 + mg cos a) Show that the system can display bounded or unbounded motion, depending on the value of the energy. Show that the motion is always bounded if E &lt;...

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612 THEORETICAL PH MECHANICS Problem Assignment # 1 Winter 2009 01/07/09 due 01/14/09 1. Pendulum (Part 1) note: This problem will be continued in later assignments. Consider a mathematical pendulum (mass m, length ) with Lagrangian L= m 2 2 2 + mg cos a) Show that the system can display bounded or unbounded motion, depending on the value of the energy. Show that the motion is always bounded if E < U0 . Determine U0 and the turning points 0 of the oscillation. b) For the case of bounded motion, nd the oscillation frequency as a function of E. Discuss your result as a function of = E/2U0 + 1/2, especially the limiting cases 0 and 1. hint: The result can be expressed in terms of the complete elliptic integral of the rst kind, K(k), dened as /2 1 K(k) = dx 0 1 k 2 sin2 x c) Show that (t) can be expressed in closed form in terms of the Jacobi elliptic function cn: (t) = 20 sin(0 /2) cn(0 t, k) , where 0 = U0 /m 2 . x hint: The elliptic integral of the rst kind, F (x, k), is dened as F (x, k) = 0 dz 1 1 k 2 sin2 z (k 1) . Its inverse is known as the amplitude, x F = (y, k) and cn is dened as cn(x, k) = cos am(x, k) . Also useful are sn(x, k) = sin am(x, k) , and dn(x, k) = d am(x, k) = dx 1 k 2 sin2 y . (15 points) .../over y = am(x, k) , 2. Geodesics on Manifolds, Rotational Invariance, and Angular Momentum Conservation Let x( ) = (x1 ( ), x2 ( ), x3 ( )) be a path in I 3 parameterized by some parameter . Then the length R of the path between the points x = x( ) and x+ = x(+ ) is given by the functional + S= d L(x, x, ) with L(x, x, ) = L(x) = x2 + x2 + x2 1 2 3 a) Show that the geodesics in I 3 , i.e. the paths whose lenghts are stationary, are given by straight R lines. Now consider the geod...

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Oregon - PHYS - 612
PH 612THEORETICAL MECHANICS Take-Home Final ExamWinter 200902/02/09 due 02/16/09Note: You may use any inanimate resource in solving the problems given below, but you must not obtain any help from, or discuss your progress with, any human being
Oregon - PHYS - 612
Oregon - PHYS - 612
Oregon - PHYS - 612
Oregon - PHYS - 612
Oregon - PHYS - 612
Some background information for Problem # 2Let S1 =tftfdt L1 (q, q, t) , tiS2 =tidt L2 (q, q, t) be two functionals. Suppose we want to nd those curves q(t) that make S1 stationary while obeying the constraint S2 = const. () This can
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blrms2frameMasahiro Ito University of Oregon 7/15/07 This program generates time series of weighted band limited RMS from frame data for multiple channels and saves them in frame les.Usage:blrms2frame -conf &lt;config file&gt; -start &lt;start gps&gt; -end &lt;
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Oregon - EC - 370
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Oregon - IWOMP - 2005
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Oregon - IWOMP - 2005
IWOMP 2005 - OMPlab on Sun1/8OMPlab on Sun - Lab ProgramsIntroduction We have prepared a set of lab programs for you. We have an exercise to implement OpenMP in a simple algorithm and also various programs to demonstrate certain features of the
Oregon - PH - 162
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Oregon - PH - 162
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STATUS REPORT Development of a Silicon-tungsten Test Module for an Electromagnetic CalorimeterPersonnel and Institution(s) requesting funding University of Oregon, Department of Physics and Oregon Center for HEP: Jim Brau (faculty), Raymond Frey (fa
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LIGO-T010049-00-W: A Report on Interferometer and Mode Cleaner Transients During the E2 Engineering RunRauha Rahkola1, Ray Frey1, Rick Savage2The TaskThe Detector Characterization group encouraged involvement in the Engineering Run of November 20
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Measuring the Lifetime of the Muon ParticleAustyn YoungQuantum Field TheorySmall Classical mechanics e.g. F = ma Fast Relativistic mechanics e.g. E = mc2 Quantum mechanics e.g. Heisenberg uncertainty principle Quantum Field Theory e.g. force carr
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