MATH0054 (Analytical Dynamics)

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MATH0054 (Analytical Dynamics) Year: 2019{2020 Code: MATH0054 Old Code: MATH7302 Level: 6 (UG) Normal student group(s): UG: Year 2 and 3 Mathematics degrees Value: 15 credits (= 7.5 ECTS credits) Term: 2 Structure: 3 hour lectures per week. Assessed coursework. Assessment: 90% examination, 10% coursework. In order to pass the module you must have at least 40% for both the examination mark and the ﬁnal weighted mark. Normal Pre-requisites: MATH0009 (previously MATH1302), MATH0011 (previously MATH1402) Lecturer: Prof A Sokal Course Description and Objectives Analytical dynamics develops Newtonian mechanics to the stage where powerful mathematical techniques can be used to determine the behaviour of many physical systems. The mathematical framework also plays a role in the formulation of modern quantum and relativity theories. Topics studied are the kinematics of frames of reference (including rotating frames), dynamics of systems of particles, Lagrangian and Hamiltonian dynamics and rigid body dynamics. The emphasis is both on the formal development of the theory and also use of theory in solving actual physical problems. Recommended Texts Relevant books are: (i) Gregory, Classical Mechanics; (ii) T L Chow, Classical Mechanics; (iii) Goldstein, Classical Mechanics; (iv) Taylor, Classical Mechanics; (v) Marion, Classical Dynamics of Particles and Systems Detailed Syllabus − Review of the fundamental principles of Newtonian mechanics; Galileo's principle of relativity. − Systems of particles and conservation laws: linear momentum, angular momentum, energy (internal and external potentials). − Systems of coupled linear oscillators: normal modes. − Introduction to perturbation theory for anharmonic oscillators. − Lagrangian dynamics: generalised coordinates, variational principles, symmetries and conservation laws. − Hamiltonian dynamics: phase space, Poisson brackets, introduction to canonical transfor- mations. − Kinematics and dynamics in noninertial reference frames: centrifugal and Coriolis pseudo- forces. − Rigid bodies: Eulerian angles, inertia matrices, Euler's equations of motions, force-free motion, tops. September 2019 MATH0054.

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All-Master-File-Problem-Set
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 THE CITY UNIVERSITY OF NEW YORK First Examination for PhD Candidates in Physics Analytical Dynamics Summer 2010 Do two of the following three problems. Start each problem on a new page. Indicate clearly which two problems you choose to solve. If you do not indicate which problems you wish to be graded, only the ﬁrst two problems will be graded. Put your identiﬁcation number on each page. 1. A particle moves without friction on the axially symmetric surface given by 1 z = br2, x = r cos φ, y = r sin φ 2 where b > 0 is constant and z is the vertical direction. The particle is subject to a homogeneous gravitational force in z-direction given by −mg where g is the gravitational acceleration. (a) (2 points) Write down the Lagrangian for the system in terms of the generalized coordinates r and φ. (b) (3 points) Write down the equations of motion. (c) (5 points) Find the Hamiltonian of the system. (d) (5 points) Assume that the particle is moving in a circular orbit at height z = a. Obtain its energy and angular momentum in terms of a, b, g. (e) (10 points) The particle in the horizontal orbit is poked downwards slightly. Obtain the frequency of oscillation about the unperturbed orbit for a very small oscillation amplitude. 2. Use relativistic dynamics to solve the following problem: A particle of rest mass m and initial velocity v0 along the x-axis is subject after t = 0 to a constant force F acting in the y-direction.
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