Lecture

Non-linear dynamics

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In course

PHYS-751: Advanced concepts in particle accelerators

Accelerator physics covers a wide range of very exciting topics. This course presents basic physics ideas and the technologies underlying the workings of modern accelerators. An overview of the new ideas and challenges of the possible paths towards the nex ...

Description

This lecture covers contemporary methods, techniques, and tools in nonlinear dynamics, emphasizing practical applications over theoretical exercises. Starting from linear dynamics, topics progress to Lagrange and Hamiltonian formalism, Lie operators, nonlinear maps, symplecticity, and beam-beam effects. The instructor motivates the study of nonlinear dynamics in accelerators, highlighting the importance of canonical variables and the transition to nonlinear elements. The lecture explores the challenges of integrating nonlinear fields, stability analysis, and the use of simulations for realistic models. Symplectic integration methods, thin lens approximations, and the symplecticity of maps are discussed in detail, providing insights into accurate and efficient computations in accelerator physics.

Instructors (3)

Tatiana Pieloni

Expertise Beam dynamics in accelerators.Colliders luminosity and beam-beam interactions.Collective effects. Teaching & PhD PhD Students Yi Wu, Lode Vanhecke Past EPFL PhD Students as codirector Xavier Buffat, Claudia Tambasco Courses Advanced concepts in particle accelerators PHYS-751 Accelerator physics covers a wide range of very exciting topics. This course presents basic physics ideas and the technologies underlying the workings of modern accelerators. An overview of the new ideas and

Rasmus Ischebeck

Teaching & PhD PhD Students Maria Paula Rey Barrera, Raziyeh Dadashi Motlagh, Evan James Ericson Past EPFL PhD Students as codirector Bennie Smit, Ishkhan Gorgisyan

Werner Friedrich Herr

Teaching & PhD Courses Advanced concepts in particle accelerators PHYS-751 Accelerator physics covers a wide range of very exciting topics. This course presents basic physics ideas and the technologies underlying the workings of modern accelerators. An overview of the new ideas and challenges of the possible paths towards the next generation of accelerators will be given.

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Related concepts (32)

Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group".

Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds.

Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Hamiltonian field theory

In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.

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.

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