BSc:TheoreticalMechanics old

Theoretical Mechanics

• Course name: Theoretical Mechanics
• Course number: XYZ
• Knowledge area: Mechanical Engineering - Fundamentals

Administrative details

• Faculty: Computer Science and Engineering
• Year of instruction: 3rd year of BS
• Semester of instruction: 1st semester
• No. of Credits: 4 ECTS
• Total workload on average: 144 hours overall
• Class lecture hours: 2 per week.
• Class tutorial hours: 2 per week.
• Lab hours: 2 per week.
• Individual lab hours: 0.
• Frequency: weekly throughout the semester.
• Grading mode: letters: A, B, C, D.

Prerequisites

• Introductory Electricity and Magnetism
• Vector Calculus
• Ordinary Differential Equations

Course outline

This course is a broad, theoretical treatment of classical mechanics, useful in its own right for treating complex dynamical problems, but essential to understanding the foundations of quantum mechanics and statistical physics.

Expected learning outcomes

• Solve problems with symbolic (rather than numeric) parameters
• Evaluate and articulate whether an answer is reasonable using limiting case analysis, dimensional analysis or multiple solutions paths
• Coordinate multiple representations (e.g. verbal/text descriptions, diagrams, algebraic equations, free-body diagrams, matrix equations, space-time diagrams, etc) to solve intermediate mechanics problems
• Use Newtonian, Lagrangian and Hamiltonian methods for solving mechanics problems
• Use Lorentz transformations to describe physical situations in inertial reference frames
• Apply conservation laws appropriately in non-relativistic and relativistic situations

Expected acquired core competences

• Determine the nature of the constraints (holonomic or non-holonomic, time-dependent or static) and forces (conservative or non-conservative) for a given problem, and thereby identify the number of degrees of freedom and select appropriate generalized coordinates.
• Obtain the Lagrangian, the generalized forces and momenta for a given problem, and the appropriate equations of motion. Solve the equations for standard problems, including small oscillation approximations. Be able to apply the concept of Lagrange multipliers to obtain the forces of constraint for a given mechanical system.
• Be able to apply mechanical gauge transformations, and treat problems in which (i) generalized potentials appear (e.g., charged particles in electromagnetic fields, dissipate forces) and (ii) forces that do not have an associated potential or the existence of V(q) is ignored.
• Obtain conserved quantities from symmetry properties of the Lagrangian using the concept of cyclic coordinates and be able to interpret the physical significance of these quantities (e.g., linear & angular momentum, energy).
• Obtain the equation of motion in terms of an effective potential and solve if for standard central force problems, e.g., orbit equation, precession, etc.
• Be able to derive a scattering cross section for a given central force problem.
• Apply the Lagrangian theory to a system of coupled oscillators and write down the equations of motion.
• Apply the concept of normal mode theory to find the normal frequencies and normal modes of the system.
• Reconstruct the motion of the system from the normal modes.
• Apply Lorentz transformation to standard problems in special relativity.
• Be able to describe particle dynamics in terms of covariant tensor equations based on Minkowski geometry.
• Recognize the invariance of the scalar product and its consequences to an extent that allows the student to compute threshold energies in the center-of-momentum and laboratory systems for particle collision/production processes. Examples include Compton scattering and p+p processes.
• Recognize the concepts of Hamilton?s principle and phase-space, be able to obtain the Hamiltonian for a problem with constraints, the resulting Hamilton?s equations of motion and discuss conserved quantities from a cyclic coordinate perspective. Be able to recognize the resulting reduction of the dimensionality of the problem (number of degrees of freedom).
• Solve Hamilton?s equations of motion for standard problems
• Be able to find the canonical transformation for simple problems to make the problem easy to solve (cyclic coordinates).
• Be able to test for the canonical condition using generating functions, the symplectic condition, and Poisson brackets.
• Apply the Poisson bracket formalism to identify constants of motions. Be able to determine the time evolution of a generic dynamic variable of interest for a given Hamiltonian using Poisson bracket formalism.
• Recognize Liouville’s theorem and be able to prove it from phase space considerations.

Detailed topics covered in the course

This course will cover concepts in classical mechanics, which are rooted in Newton?s Laws of motion. We will cover new techniques, in particular the Lagrangian and Hamiltonian formulations, that allow the analysis of systems for which a direct application of Newton?s Laws would be cumbersome. These concepts will be applied to the motion of oscillators, rigid bodies (in particular their rotation), and particles moving in a central potential (gravity being one example).

Textbook

• Classical Dynamics of Particles and Systems 5th Edition Stephen T. Thornton | Jerry B. Marion

Required computer resources

No special needs.

Evaluation

• Weekly quizzes (20%)
• Home assignments (30%)
• Midterm Exam (20%)
• Final Exam (30%)