Module Specifications.
Current Academic Year 2024 - 2025
All Module information is indicative, and this portal is an interim interface pending the full upgrade of Coursebuilder and subsequent integration to the new DCU Student Information System (DCU Key).
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Date posted: September 2024
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Repeat examination |
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Description The purpose of this module is for the student to demonstrate knowledge and understanding in fundamental concepts and calculation methods of dynamics of system of particles. The Newtonian, Lagrangian and Hamiltonian formulation of mechanics will be taught. The ultimate goal the derivation (and its study) of the equations of motion for elementary mechanical systems using all the above-mentioned formulations. A secondary goal is a demonstration of mathematical methods as applied to physics. | |||||||||||||||||||||||||||||||||||||||||||
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Learning Outcomes 1. Define and understand basic concepts related to mechanical systems of particles 2. Describe and understand the three most important methodological approaches of Classical Mechanics (Newtonian, Lagrangian, Hamiltonian) 3. Apply classical dynamics methods to fundamental problems of classical mechanics (e.g. harmonic oscillators, Kepler's laws, particle collisions) 4. Understand and apply differential and integral analysis of functions (e.g. ordinary differential equations, function minimization) in physical (space-time) problems | |||||||||||||||||||||||||||||||||||||||||||
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml |
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Indicative Content and Learning Activities
Lecture Series: Newtonian Dynamics I - momentum and forceBasics of kinematics in 1-2 and 3-D motion; position, velocity and acceleration. Momentum, force and Newton's 1st and 2nd laws of motion. Examples for constant (~c) linear (~x), quadratic (~x^2) and inverse square (~1/x^2) forces; free fall, charge in an constant electric field, Hooke's law, Coulomb's electrostatic force. Conservation of momentum, multiparticle systems. Conservation of momentum, Newton's 3rd law of motion.Lecture Series: Newtonian Dynamics II - EnergyKinetic and potential energy and Newton's laws; Work and conservation of mechanical energy; Derivation of equations-of-motion from the conservation of energy theorem; Electric and gravitational potential energy, harmonic oscillator (mass-spring and pendulum); Kinetic and potential energy of multiparticle systems, Internal potential energy of a system. Non-conservative forces and friction.Lecture Series: Newtonian Dynamics IIIMotion in 2- and 3-D space; angular momentum and angular momentum conservation law. Kinetic, potential energy and planar motion in a central force field; Coulomb's (electrostatic) Newton's (gravitational) laws; Damped and driven harmonic oscillator; physical pendulum, projectile motion, Kepler's planetary motion laws - Rutherford's scattering lawLecture Series: Lagrangian method to 1-D systemsLagrangian and the principle of least-action (Hamilton principle); the action integral Euler-Lagrange equations, generalized coordinates and momentum, constrained systems. Examples for free-fall/projectile motions, harmonic oscillator, pendulum problems; pulley motion Lagrangian of Multiparticle systems, the case of coupled harmonic oscillators.Lecture Series: Hamilton's method to 1-D systemsHamilton’s equations of motion for simple one-dimensional systems; Phase space, conservative systems and conservation of energy, the Hamiltonian of simple physical systems. Motion in phase space. Phase space diagrams. Connection with quantum mechanics.Learning Activities:Tutorial problems are worked out and given for homework. Physics: tutorial problems for homework with subjects relevant to the contents above. Most problems are asked to be solved using all three formulations of mechanics (Newton's, Lagrange's, Hamilton's). Maths: Cartesian and Polar coordinate systems. Vector algebra, line integral, ordinary differential equations with constant coefficients. Basic single-variable functional calculus, functions and functional minimization, Lagrange multipliers. Elements of differential and integral calculus of functions (ordinary differential equations and variational method of function minimization) | |||||||||||||||||||||||||||||||||||||||||||
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Indicative Reading List
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Other Resources None | |||||||||||||||||||||||||||||||||||||||||||