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Semester 1

Lagrangian Dynamics (PHYS10015)


Physics and Astronomy





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Course Summary

The principles of classical dynamics, in the Newtonian formulation, are expressed in terms of (vectorial) equations of motion. These principles are recapitulated and extended to cover systems of many particles. The laws of dynamics are then reformulated in the Lagrangian framework, in which a scalar quantity (the Lagrangian) takes centre stage. The equations of motion then follow by differentiation, and can be obtained directly in terms of whatever generalised coordinates suit the problem at hand. These ideas are encapsulated in Hamilton's principle, a statement that the motion of any classical system is such as to extremise the value of a certain integral. The laws of mechanics are then obtained by a method known as the calculus of variations. As a problem-solving tool, the Lagrangian approach is especially useful in dealing with constrained systems, including (for example) rotating rigid bodies, and one aim of the course is to gain proficiency in such methods. At the same time, we examine the conceptual content of the theory, which reveals the deep connection between symmetries and conservation laws in physics. Hamilton's formulation of classical dynamics (Hamiltonian Dynamics) is introduced, and some of its consequences and applications are explored.

Course Description

- Revision of Newtonian Mechanics: Newton's laws; Dynamics of a Particle; Conservation Laws- Dynamics of a system of particles; Momentum; Angular Momentum; Energy; Transformation Laws- Use of centre of momentum; Noninertial rotating frames; Summary of Newton's scheme- Constraints; Generalised coordinates and velocities- Generalised forces; Derivation of the Lagrange equation- Lagrangian; Examples- Using Lagrangian Method. Examples: Atwood's Monkey; particle and wedge; simple pendulum; spherical pendulum- Rotating frames; Calculus of Variations- Applications of Variational Calculus; Hamilton's Principle- Hamilton's Principle; Conservation Laws; Energy Function - Energy Function; Conservation Laws and Symmetry - Velocity-dependent Forces; - Hamiltonian Dynamics; relationship to Quantum Mechanics - Rigid Body Motion; Introduction; Euler's Equations - The Symmetric Top - Precession; the Tennis Racquet Theorem - Lagrangian for a Top; Equations of motion; Conservation Laws - Symmetric Tops: Zones; Steady Precession; Nutation; Gyroscopes - Small Oscillation Theory

Assessment Information

Written Exam 100%, Coursework 0%, Practical Exam 0%

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