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 problemsolving 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
 Velocitydependent 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

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2023/24, 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 
Optional handins fortnightly 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)   2:00  

Academic year 2023/24, Partyear 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 
Optional handins fortnightly 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)  Semester 1 Visiting Students Only  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.

Additional Information
Graduate Attributes and Skills 
Not entered 
Additional Class Delivery Information 
Workshop/tutorial sessions, as arranged. 
Keywords  LagD 
Contacts
Course organiser  Dr Jennifer Smillie
Tel: (0131 6)50 5239
Email: J.M.Smillie@ed.ac.uk 
Course secretary  Mrs Jo Thorpe
Tel:
Email: jthorpe3@ed.ac.uk 

