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Lecture

Hamiltonian Mechanics: Phase Space and Poisson Bracket

In course

CH-453: Molecular quantum dynamics

The course covers several exact, approximate, and numerical methods to solve the time-dependent molecular Schrödinger equation, and applications including calculations of molecular electronic spectra. More advanced topics include introduction to the semicl ...

Description

This lecture covers the concepts of phase space, the Ehrenfest theorem, and the Poisson bracket in the context of Hamiltonian mechanics. It discusses the angular momentum, momentum, and Hamiltonian functions, emphasizing the time dependence of the Hamiltonian. The lecture also explores the derivation of Hamilton's equations from the variational principle and the connections between quantum and classical mechanics through the Ehrenfest theorem. Additionally, it delves into the Poisson algebra and Lie algebra, highlighting the bilinearity and symmetry properties of these mathematical structures.

Instructor

Jiri Vanicek

Official source

Ontological neighbourhood

Physics

Mechanics: Classical mechanics

Related concepts (32)

Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).

Variational principle

In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems.

Hamiltonian mechanics

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.

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