Analytical Mechanics

Knowledge:

The student will be trained in the ability to solve a problem by many different methods, some of 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 judicious choices of generalised 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 description of a given physical problem.

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 generalised 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 examples.
• 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.

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 mechanics should rightly been seen as the true formal basis. As such, this course will give the student the needed overview and common ground of all fundamental physical laws.