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Tune Shift With Amplitude
The tune shift with amplitude is an important concept in circular accelerators or synchrotrons. The machine may be described via a symplectic one turn map at each position, which may be thought of as the Poincaire section of the dynamics. A simple harmonic oscillator has a constant tune for all initial positions in phase space. Adding some non-linearity results in a variation of the tune with amplitude. Amplitude may refer to either the initial position, or more formally, the initial action of the particle. Definition Consider dynamics in phase space. These dynamics are assumed to be determined by a Hamiltonian, or a symplectic map. For each initial position, we follow the particle as it traces out its orbit. After transformation into action-angle coordinates, one compute the tune \nu and the action J. The tune shift with amplitude is then given by \frac. The transformation to action-angle variables out of which the tune may be derived may be considered as a transformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Accelerator
A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental research in particle physics. Accelerators are also used as synchrotron light sources for the study of condensed matter physics. Smaller particle accelerators are used in a wide variety of applications, including particle therapy for oncology, oncological purposes, Isotopes in medicine, radioisotope production for medical diagnostics, Ion implantation, ion implanters for the manufacturing of Semiconductor, semiconductors, and Accelerator mass spectrometry, accelerator mass spectrometers for measurements of rare isotopes such as radiocarbon. Large accelerators include the Relativistic Heavy Ion Collider at Brookhaven National Laboratory in New York, and the largest accelerator, the Large Hadron Collider near Geneva, Switzerland, operated b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synchrotron
A synchrotron is a particular type of cyclic particle accelerator, descended from the cyclotron, in which the accelerating particle beam travels around a fixed closed-loop path. The strength of the magnetic field which bends the particle beam into its closed path increases with time during the accelerating process, being ''synchronized'' to the increasing kinetic energy of the particles. The synchrotron is one of the first accelerator concepts to enable the construction of large-scale facilities, since bending, beam focusing and acceleration can be separated into different components. The most powerful modern particle accelerators use versions of the synchrotron design. The largest synchrotron-type accelerator, also the largest particle accelerator in the world, is the Large Hadron Collider (LHC) near Geneva, Switzerland, completed in 2008 by the European Organization for Nuclear Research (CERN). It can accelerate beams of protons to an energy of 7 electron volt, teraelectro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Principles In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation. Formal definition A diffeomorphism between two symplectic manifolds f: (M,\omega) \rightarrow (N,\omega') is called a symplectomorphism if :f^*\omega'=\omega, where f^* is the pullback of f. The symplectic diffeomorphisms from M to M are a (pseudo-)group, called the symplectomorphism group (see below). The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field X \in \Gamma^(TM) is called symplectic if :\mathcal_X\omega=0. Also, X is symplectic if the flow \phi_t: M\rightarrow M of X is a symplectomorphism for every t. These vector fields build a Lie subalgebra of \Gamma^(TM). Here, \Gamma^(TM) is the set of smooth vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action-angle Coordinates
In classical mechanics, action-angle variables are a set of canonical coordinates that are useful in characterizing the nature of commuting flows in integrable systems when the conserved energy level set is compact, and the commuting flows are complete. Action-angle variables are also important in obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. They only exist, providing a key characterization of the dynamics, when the system is completely integrable, i.e., the number of independent Poisson commuting invariants is maximal and the conserved energy surface is compact. This is usually of practical calculational value when the Hamilton–Jacobi equation is completely separable, and the separation constants can be solved for, as functions on the phase space. Action-angle variables define a foliation by invariant Lagrangian tori because the flows induced by the Poisson commuting invariants remain within their joint level sets, while ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action close to the Planck constant, quantum effects are significant. In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy. More formally, action is a mathematical functional which takes the trajectory ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Form (mathematics)
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp space, Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff convergence, Gromov–Hausdorff distance. In dynamical systems, an orbit (dynamics), orbit is called ''Lyapunov stability, Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Aperture (accelerator Physics)
The dynamic aperture is the stability region of phase space in a circular accelerator. For hadrons In the case of protons or heavy ion accelerators, (or synchrotrons, or storage rings), there is minimal radiation, and hence the dynamics is symplectic. For long term stability, tiny dynamical diffusion (or Arnold diffusion In applied mathematics, Arnold diffusion is the phenomenon of instability of nearly-integrable systems, integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More ...) can lead an initially stable orbit slowly into an unstable region. This makes the dynamic aperture problem particularly challenging. One may be considering stability over billions of turns. A scaling law for Dynamic aperture vs. number of turns has been proposed by Giovannozzi. For electrons For the case of electrons, the electrons will radiate which causes a damping effect. This means that one typically only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved Scientific Revolution, substantial change in the methods and philosophy of physics. The qualifier ''classical'' distinguishes this type of mechanics from physics developed after the History of physics#20th century: birth of modern physics, revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics. The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton, and the mathematical methods invented by Newton, Gottfried Wilhelm Leibniz, Leonhard Euler and others to describe the motion of Physical body, bodies under the influence of forces. Later, methods bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pendulum
A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing. Pendulums were widely used in early mechanical clocks for timekeeping. The regular motion of pendulums was used for timekeeping and was the world's most accurate timekeeping technology until the 1930s. The pendulum clock invented by Christiaan Huygens in 1656 became the world's standard timekeeper, used in homes and offices for 270 years, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sextupole Magnet
file:Aust.-Synchrotron,-Sextupole-Focusing-Magnet,-14.06.2007.jpg, 250px, Sextupole electromagnet as used within the storage ring of the Australian Synchrotron to correct chromatic aberrations of the electron beam file:magnetic field of an idealized sextupole.svg, 250px, Field lines of an idealized sextupole magnet in the plane transverse to the beam direction A sextupole magnet (also known as a hexapole magnet) consist of six magnetic poles set out in an arrangement of alternating north and south poles arranged around an axis. They are used in particle accelerators for the control of chromatic aberrations and for damping the head—tail transverse plasma instability. Two sets of sextupole magnets are used in transmission electron microscopy, transmission electron microscopes to correct for spherical aberration. The design of sextupoles using electromagnets generally involves six steel pole tips of alternating polarity. The steel is magnetised by a large electric current that flow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
