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The Lagrangian and Hamiltonian formulations of classical mechanics which provide essential preparation for all advanced courses in theoretical physics. Techniques learned have wide use in advanced quantum mechanics, quantum field theory, general relatively, particle physics and statistical mechanics.
In this course students will embark on a voyage of discovery of the deep theoretical principlesthat underlie Newtonian and relativistic mechanics, and to appreciate why the lawsof physics are the way they are. They will learn new ways of thinking about the physicalworld which allow deeper appreciation of the links between the classical and quantumregimes.Armed with the powerful techniques of Lagrangian and Hamiltonian dynamics, and Cartesiantensors, students will have the tools to simplify complex mechanical problems to theirbasic elements. With elegant symmetry principles such as Noether’s theorem they willunderstand the deep connection between symmetries of spacetime and conservation laws,seeing how, for example, Kepler’s second law follows from rotational symmetry and conservationof angular momentum. They will apply this new understanding to a variety ofphysical systems, from coupled oscillators to particles moving in electromagnetic fields.Finally they will discover how the symmetries of special relativity are most succinctly describedwith the language of 4-vectors, and derive the Lorentz group from the Principle ofRelativity.This course is the basis for all advanced courses in theoretical physics.OUTLINE* Dynamical systems – definitions. Constrained systems. Lagrange’s equations.* Principle of least action. Euler-Lagrange equations.* Symmetries, conservation laws and Lie groups. Noether’s theorem.* Oscillations: linearization. The linear chain.* Hamiltonian formulation. Legendre’s transformation.* Transformation theory. Canonical transformations. Generating functions. Poissonbrackets.* Hamilton-Jacobi method. Physical applications: (e.g. wave mechanics and Schr¨odinger’sequation).* Special relativity: Kinematics, symmetries and Lagrangian formulation
(1) PHYS202 orPHYS205; (2) PHYS203; (3) MATH201 RP: MATH202 and MATH203
MATH202 and MATH203
Goldstein, Herbert , Poole, Charles P., Safko, John L;
Addison Wesley, 2002.
Arnold, V. I;
Mathematical methods of classical mechanics;
I. Percival and D. Richards,;
Introduction to Dynamics;
Cambridge University Press, 1982.
L. Landau and E. Lifschitz,;
Saletan, Eugene J. , Cromer, Alan H;
Wiley, New York,, 1995.
Additional readingD.E. Bourne and P.C. Kendall, Vector Analysis and Cartesian Tensors, 3rd ed, (CRC Press 2002), chapter 8, [for Orthogonal Transformations in §3 only].D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 4th ed. (OxfordUniversity Press, 2007) chapters 1,2 [for §4. Oscillations only]N.A. Doughty, Lagrangian Interaction, (Addison Wesley, Sydney, 1990), chapters 12,13[for §6. Special Relativity only].
Course information and content (PDF 2169KB)
General Course Information (PDF 163KB)
Domestic fee $900.00
International fee $4,250.00
* All fees are inclusive of NZ GST or any equivalent overseas tax, and do not include any programme level discount or additional course-related expenses.
For further information see
School of Physical & Chemical Sciences on the department and colleges page.