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Unit of study_

MATH4077: Lagrangian and Hamiltonian Dynamics

Lagrangian and Hamiltonian dynamics are reformulations of classical Newtonian mechanics into a mathematically sophisticated framework using arbitrary coordinate systems. This formulation of classical mechanics 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 simplify apparently complicated dynamical problems. Connections between geometry and different physical theories beyond classical mechanics are explored. Students will be expected to describe and solve mechanical systems of some complexity including planetary motion and to investigate stability. Hamilton-Jacobi theory will be used to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. Students will study an application of Lagrangian and Hamiltonian dynamics described in a modern research paper.

Code MATH4077
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(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)]
Assumed knowledge:
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.