Undergraduate Course: Lagrangian Dynamics (PHYS10015)

Course Outline
School	School of Physics and Astronomy	College	College of Science and Engineering
Credit level (Normal year taken)	SCQF Level 10 (Year 3 Undergraduate)	Availability	Available to all students
SCQF Credits	10	ECTS Credits	5
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

Entry Requirements (not applicable to Visiting Students)
Pre-requisites		Co-requisites
Prohibited Combinations		Other requirements	Students should have achieved the learning outcomes of Dynamics (PHYS08040), together with knowledge of basic algebra and calculus corresponding to material in Mathematics for Physics 1 (PHYS08035), linear algebra and multivariate calculus corresponding to the material in Algebra and Calculus (PHYS08041), or their equivalents.

Information for Visiting Students
Pre-requisites	None

Course Delivery Information

Academic year 2015/16, Available to all students (SV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 20, Summative Assessment Hours 2, Revision Session Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 52 )
Assessment (Further Info)	Written Exam 100 %, Coursework 0 %, Practical Exam 0 %
Additional Information (Assessment)	Degree Examination, 100%
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S2 (April/May)			2:00

Academic year 2015/16, Part-year visiting students only (VV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 20, Summative Assessment Hours 2, Revision Session Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 52 )
Assessment (Further Info)	Written Exam 100 %, Coursework 0 %, Practical Exam 0 %
Additional Information (Assessment)	Degree Examination, 100%
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)			2:00

Learning Outcomes
Consolidation of the learning outcomes in the Entry Requirements in the context of more challenging classical dynamics problems, together with at least two of the following: a. understanding of the Lagrangian formulation of classical dynamics and the ability to apply it to solve for the motion of point particles and simple bodies in terms of generalised coordinates; b. understanding of the relationship between symmetries and conservation laws, and knowledge of the Hamiltonian formulation of classical dynamics and Poisson brackets; c. ability to apply the calculus of variations to solve minimisation problems, and knowledge of the formulation of dynamics in terms of a variational principle; d. ability to apply Lagrangian methods to solve for the motion of rigid bodies; e. ability to solve for the small amplitude oscillations of coupled systems.