Undergraduate Course: Lagrangian Dynamics (PHYS10015)
Course Outline
| School | School of Physics and Astronomy |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 10 (Year 3 Undergraduate) |
Credits | 10 |
| Home subject area | Undergraduate (School of Physics and Astronomy) |
Other subject area | None |
| Course website |
http://www2.ph.ed.ac.uk/~amorozov/teaching.html |
Taught in Gaelic? | No |
| Course description | 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. |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | Yes |
Course Delivery Information
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| Delivery period: 2012/13 Semester 2, Available to all students (SV1)
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WebCT enabled: No |
Quota: None |
| Location |
Activity |
Description |
Weeks |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
| King's Buildings | Lecture | | 1-11 | | 11:10 - 12:00 | | | | | King's Buildings | Lecture | | 1-11 | | | | | 11:10 - 12:00 | | King's Buildings | Tutorial | | 1-11 | | | | 15:00 - 17:00 | |
| First Class |
First class information not currently available |
| Additional information |
Workshop/tutorial sessions, as arranged. |
| Exam Information |
| Exam Diet |
Paper Name |
Hours:Minutes |
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| Main Exam Diet S2 (April/May) | | 2:00 | | |
Summary of Intended Learning Outcomes
On successful completion of this course a student will be able to:
1)Understand the Newtonian dynamics of a system of particles
2)Understand virtual displacements, constraints, generalised coordinates/ velocities / forces / momenta; discuss the derivation of the Euler Lagrange equations using virtual displacements
3)Apply the Lagrangian technique to solve a large range of problems in dynamics (the ethereal expression "a large range of problems" may be disambiguated by reference to the tutorial sheets and previous examinations)
4)Understand and apply the calculus of variations, discuss the derivation of the Euler Lagrange equations for constrained systems and thus appreciate Hamilton's principle as the embodiment of Lagrangian dynamics
5)Understand all of: ignorable coordinates / the origin of conservation laws, Lagrangian for a charged particle in an EM field, canonical versus mechanical momentum, allowed changes in the Lagrangian
6)Appreciate the Lagrangian for a relativistic charged particle
7)Derive the conservation of linear (angular) momentum from the homogeneity (isotropy) of space, appreciate the relation between symmetries and conservation laws
8)Define the Hamiltonian by Legendre transformation, derive Hamilton's equations of motion and apply them to simple problems; define / evaluate Poisson brackets, appreciate the connection with Quantum Mechanics
9)Understand rotating frames and the Eulerian approach to rigid body motion, and analyse torque-free motion; understand the Lagrangian formulation of the symmetric top, derive the equations of motion and conservation laws, understand nutation, precession and sleeping
10)Understand and apply small oscillation theory in the Lagrangian formulation
11)Apply all of the above to unseen problems in each formulation of classical dynamics |
Assessment Information
| Degree Examination, 100% |
Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
&· Introduction
&· 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; Lorentz Force; Special Relativity
&· Special Relativity; Hamiltonian Dynamics; 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
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| Transferable skills |
Not entered |
| Reading list |
Not entered |
| Study Abroad |
Not entered |
| Study Pattern |
Not entered |
| Keywords | LagD |
Contacts
| Course organiser | Dr Alexander Morozov
Tel: (0131 6)50 5289
Email: |
Course secretary | Miss Laura Gonzalez-Rienda
Tel: (0131 6)51 7067
Email: |
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