Unit of study_

# MATH3977: Lagrangian and Hamiltonian Dynamics (Adv)

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.

Code MATH3977 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? A mark of 65 or greater in 12 credit points of MATH2XXX units of study None MATH4077

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 a broad range of familiar 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 familiar 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 1 week before the first day of teaching for the relevant session.