Canonical Transformations: Properties and Applications

This lecture covers the concept of canonical transformations, focusing on their properties and applications in Hamiltonian mechanics. The instructor explains how these transformations preserve certain quantities and discusses their importance in describing the dynamics of physical systems. The lecture also delves into the mathematical aspects of canonical transformations, such as the Jacobian matrix and the symplectic group. Through examples and theoretical explanations, students learn how canonical transformations play a crucial role in simplifying the analysis of complex systems and understanding the evolution of dynamical systems.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (41)

Related concepts (39)

PHYS-202: Analytical mechanics (for SPH)

Présentation des méthodes de la mécanique analytique (équations de Lagrange et de Hamilton) et introduction aux notions de modes normaux et de stabilité.

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.

Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Hamiltonian system

A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system.

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.

Dynamical systems theory

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.

Canonical Transformations in Analytical Mechanics PHYS-202: Analytical mechanics (for SPH)

Explores canonical transformations, conservation of quantities, and differential equations in analytical mechanics.

Canonical Transformations: Symplectic Group and Properties PHYS-202: Analytical mechanics (for SPH)

Explores canonical transformations, symplectic groups, real matrices, preserved quantities, and volumes in phase space.

Monstrous Moonshine MATH-680: Monstrous moonshine

Explores Monstrous Moonshine, focusing on the 1979 discovery and its mathematical connections.

Canonical Transformations 2 PHYS-202: Analytical mechanics (for SPH)

Explores canonical transformations, symplectic matrices, and the concept of identity in matrix space.

Canonical Equations and Integrable Systems PHYS-202: Analytical mechanics (for SPH)

Explores canonical equations, integrable systems, trajectories, and the symplectic matrix in understanding system dynamics.