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During 2021 we will continue to support students who need to study remotely due to the ongoing impacts of COVID-19 and travel restrictions. Make sure you check the location code when selecting a unit outline or choosing your units of study in Sydney Student. Find out more about what these codes mean. Both remote and on-campus locations have the same learning activities and assessments, however teaching staff may vary. More information about face-to-face teaching and assessment arrangements for each unit will be provided on Canvas.

Unit of study_

MATH4077: Lagrangian and Hamiltonian Dynamics

Lagrangian and Hamiltonian dynamics are a reformulation of classical Newtonian mechanics into a mathematically sophisticated framework that can be applied in many different coordinate systems. This formulation generalises elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamics from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear simpler. In this unit you will also explore connections between geometry and different physical theories beyond classical mechanics. You will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. You will use Hamilton-Jacobi theory to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.

Code MATH4077
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites:
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(A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 orMATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3978 or MATH3979)]
Corequisites:
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None
Prohibitions:
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MATH3977
Assumed knowledge:
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6cp of 1000 level calculus units and 3cp of 1000 level linear algebra and (MATH2X21 or MATH2X61)

At the completion of this unit, you should be able to:

  • LO1. Recall and explain fundamental definitions, equations and techniques of Lagrangian and Hamiltonian dynamics and the calculus of variations.
  • LO2. Predict essential properties of the motion in a central force field.
  • LO3. Create descriptions of new mechanical systems using Euler-Lagrange equations and analyse and describe the motion determined by these equations.
  • LO4. Explain the concept of a point transformation and apply these in novel contexts.
  • LO5. Design sets of coordinates that are adapted to describe a particular mechanical system.
  • LO6. Analyse systems with constraints using the Lagrangian approach.
  • LO7. Simplify dynamical problems by using context-dependent approaches including applying the relationships between conservation laws and symmetries or separation of variables.
  • LO8. Solve separable dynamical systems with Hamilton-Jacobi theory.
  • LO9. Verify that a given transformation is canonical and produce examples of canonical transformations using generating functions. Apply the Poisson bracket.
  • LO10. Understand the concept of integrable Hamiltonian system and find action variables.

Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.