Analytical Mechanics

Course content

This course addresses itself to all who wish to understand the more formal aspects of Physics, and in particular it is a "must" for anybody wishing to specialize in Theoretical Physics. The course introduces the mathematical formalism that underlies all of the fundamental physical laws in classical physics, and which, most remarkably, also forms the fundation of Quantum Mechanics and Quantum Field Theory.

Analytical Mechanics is fascinating because its starting point is just Newton's three laws of mechanics. From this one can, step by step, arrive at a formalism at a far deeper level. This turns out to lead to the common foundation upon which all fundamental theories of Physics stand. In detail, the course arrives there by first systematically exploring consequences of Newtonian mechanics with constraints. One is naturally led to what is known as the Lagrangian description of classical mechanics in terms of generalized coordinates, and, in a beautiful jump, into a description based on both generalized coordinates and generalized momenta in what is called the Hamiltonian formalism.

This apparent doubling of degrees of freedom leads to the notion of canonical variables, upon which Quantum Mechanics is built.

Analytical Mechanics also allows us to describe systems with a continuous number of degrees of freedom, as in fluid dynamics, the description of membranes, etc. As such, its extension to Quantum Mechanics directly points towards Quantum Field Theory.

Education

BSc Programme in Physics

 
Learning outcome

Skills:

After finishing this course the student is expected to be able to

  • Write down and apply to specific problems the equations of motion in accelerated coordinate systems, including the use of centrifugal force and Coriolis forces.
  • Select relevant generalized coordinates and establish the Lagrangian and associated equations of motion for mechanical systems with constraints.
  • Derive and use the Euler-Lagrange variational principle, including the use of Lagrange multipliers.
  • Describe and use a Legendre transform for mechanical systems.
  • Derive and use the Hamiltonian formulation of mechanics, including its use in specific examples.
  • Describe the notion of canonical transformations and apply such transformations in specific xamples.
  • Express the Hamiltonian formulation in terms of Poisson brackets and use Poisson brackets in practical examples.
  • Establish Hamilton-Jacobi and be able to use that formulation to solve mechanical problems.
  • Derive Euler-Lagrange equations of motion for a continuous system such as the vibrating string and understand the connection to Sturm-Liouville theory.

 

Knowledge:

The student will be trained in the ability to solve a problem by many different methods, some which may be much simpler than others. The student will understand the relation between symmetries and conserved quantities, concepts known and treated in classical physics which survive in Quantum Mechanics. Many problems can be solved exactly, often by judiciuous choices of generalized coordinates and momenta, others will be only approximations to the full solution, typically valid in certain domains. Crucial in this respect is the ability to find the most appropriate desciption of a given physical problem.

Competences:

The overarching goal of this course is to formulate classical mechanics at the deeper level of the Lagrange and Hamiltonian formalisms. These more fundamental formulations allow us to solve problems that would be intractable by standard Newtonian methods, and they lead us directly to the formulation of Quantum Mechanics. The origin of these reformulations of classical was at the practical level, allowing for the solution of problems in mechanics, astronomy, fluid mechanics and so on, but these reformulations of classical mecahnics should rightly been seen as the true formal basis. As such, this course will give the student the needed overview and common griound of all fundamental physical laws.

lectures and exercises

See Absalon for final course material. The following is an example of expected course material.

 

“Theoretical Mechanics of Particles and Continua"
AUTHOR:  Alexander L. Fetter, John Dirk Walecka
ISBN:  0486432610

corresponding to the mandatory physics and mathematics courses of the first year of the Physics B.Sc.

The course is identical to NFYB14006U Analytisk mekanik
It is not allowed to pass both courses.

Individual

The students receive feedback on the assignments that count towards the final grade.

ECTS
7,5 ECTS
Type of assessment
Written examination, 4 hours under invigilation
Continuous assessment
4 hour written exam Counts for 85% of the final grade.
3 homework assignments Count for 15% of the final grade.
These elements do not have to be passed separately.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
several internal examiners
Criteria for exam assessment

see learning outcome

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 28
  • Preparation
  • 142
  • Theory exercises
  • 32
  • Exam
  • 4
  • English
  • 206