Application: Kinematical and dynamical symmetries - Lec 28 - Frederic Schuller

The lecture begins with a review of classical mechanics and introduces the concept of kinematical and dynamical symmetries. The distinction between these symmetries is explored from the perspective of the cotangent bundle. The role of observables as smooth functions on a symplectic manifold is discussed, and the importance of Darboux coordinates in revealing the simplicity of the symplectic form is highlighted.

The paragraph delves into the definition of a Hamiltonian vector field, which is associated with every observable on a symplectic manifold, not just the Hamiltonian. It explains how these vector fields are derived from the exterior derivative of an observable function and how they relate to the geometric structure of the manifold. The concept of level sets of an observable and how they interact with the Hamiltonian vector field is also discussed.

This section focuses on visualizing the Hamiltonian vector field in the context of an observable's level sets. It describes the process of calculating the gradient of a function and how it gives rise to a covector field, which is then transformed using the symplectic form's inverse. The resulting vector field is orthogonal to the level sets, providing a geometric interpretation of the Hamiltonian dynamics.

The lecture introduces Poisson brackets, which are bilinear maps defined for smooth functions on the symplectic manifold. It discusses the properties of Poisson brackets, including their anti-symmetry and the Jacobi identity, which is essential for proving the algebraic structure of the observables. The connection between Hamiltonian vector fields and Lie algebra is also established.

The paragraph explains the concept of a dynamical system in classical mechanics, consisting of a symplectic manifold, a Hamiltonian, and how the change of an observable over time is defined. It also discusses the definition of a conserved quantity, which is an observable with a vanishing Poisson bracket with the Hamiltonian. The importance of the Liouville operator and its role in the formal solution of Hamilton's equations is highlighted.

The lecture distinguishes between kinematical and dynamical symmetries, with kinematical symmetries referring to the geometry of the system without forces, such as straight-line motion. Dynamical symmetries, on the other hand, are related to the potential and forces acting on the system. The conditions for a symmetry to be kinematical are explored, including the requirement for the push forward of the Hamiltonian vector field to exist and be a Killing vector field.

This section discusses how the concepts of kinematical and dynamical symmetries apply to Hamiltonian systems. It provides examples of each type of symmetry, such as angular momentum for kinematical and the Runge-Lenz vector for dynamical. The importance of the Killing equation and the role of Killing tensors in understanding dynamical symmetries are also covered.

The paragraph outlines an algorithm for finding symmetries of a Hamiltonian system, based on the assumption that a conserved quantity can be expanded in a series involving the momenta. It details the process of deriving a set of conditions that must be satisfied for each order in the momentum, leading to the generalized Killing condition for conserved quantities.

The lecture uses angular momentum as an example to illustrate the conditions for a quantity to be a conserved quantity and a kinematical symmetry. It shows that the angular momentum components satisfy the stopping condition for the series expansion and are thus conserved. The requirement for the potential to be invariant under rotations for the angular momentum to be conserved is also discussed.

The final paragraph summarizes the conditions for a quantity to be considered a kinematical symmetry. It emphasizes that only first-order terms in the momentum have the potential to be kinematical symmetries, and any dependence on the momentum higher than the first order indicates a dynamical symmetry. The lecture concludes with a thank you to the audience.