Advanced mechanics

The student is expected to understand and be able to work with the following aspects of classical mechanics after completing the course:

* The Hamilton and Lagrange formalism and how these relate to each other, as well as the concepts of potential and kinetic energy, d’Alembert’s principle, and phase space. The student can derive equations of motion from any of these formalisms for (simple many-body) particle systems, which they can solve analytically.

* Lagrange multipliers for imposing restrictions on the motion of objects. The student can apply these to obtain (coupled) differential equations for the dynamics in constrained systems. They are also familiar with the chaotic nature of the double pendulum and basic concepts that characterize chaotic dynamics, such as Lyapunov exponents.

* Hamilton’s variational principle and Noether’s theorem, which relates symmetries in Lagrangian to conserved quantities. The student can formulate an action integral and derive equations of motion / conserved quantities from this.

* Center of mass, moments of inertia, angular momentum and kinetic energy (orbital and rotational) of rigid bodies with respect to an arbitrary axis. The student can determine the principal axes and analyze the free and forced rotation of solid bodies (precession, nutation).

* Dynamics in non-inertial systems. The student can determine equations of motion in such systems and identify the centrifugal and Coriolis forces, if present.

* Normal modes of complex vibrating systems, such as coupled harmonic oscillators. The student can determine these modes and understands how they can be used to explain the behavior of physical systems, e.g., crystals.

* The basics of symplecticity, manifolds, and algebras, and their significance for the way one can study classical dynamics using geometry and algebra.