# Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. The Lagrangian is named after Italian-French mathematician and astronomer Joseph Louis Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Lagrange known as Lagrangian mechanics. Definition In classical mechanics, the natural form of the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V.[1] In symbols, If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation. , but solving any equivalent Lagrangians will give the same equations of motion.[2][3] The Lagrangian formulation Simple example The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum. Importance does not depend on . . .

Linear dynamical system Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. Introduction In a linear dynamical system, the variation of a state vector (an -dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by varies continuously with time or as a mapping, in which varies in discrete steps These equations are linear in the following sense: if and are two valid solutions, then so is any linear combination of the two solutions, e.g where need not be symmetric. Solution of linear dynamical systems If the initial vector is aligned with a right eigenvector If ) of the matrix .

Scalar field theory In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field. The quanta of the quantized scalar field are spin-zero particles, and as such are bosons. The only fundamental scalar field that has been observed in nature is the Higgs field. , has a particularly simple form: it is diagonal, and here we use the + − − − sign convention. Classical scalar field theory Linear (free) theory where is known as a Lagrangian density, dD-1 ≝ dx⋅dy⋅dz ≝ dx1⋅dx2⋅dx3 for the three spatial coordinates, is the Kronecker delta function and where for the ρ-th coordinate xρ . . is sometimes known as a mass term, due to its interpretation in the quantized version of this theory in terms of particle mass. The equation of motion for this theory is obtained by extremizing the action above. is the Laplace operator. The n! . , of to be satisfies .

Limit-cycle Stable limit cycle (shown in bold) and two other trajectories spiraling into it In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real world oscillatory systems. Definition We consider a two-dimensional dynamical system of the form where is a smooth function. with values in which satisfies this differential equation. such that for all . Properties By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve. Stable, unstable and semi-stable limit cycles Stable limit cycles are examples of attractors. Finding limit cycles . Open problems See also

Chiral symmetry breaking In particle physics, chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of a gauge theory such as Quantum Chromodynamics, the quantum field theory of the strong interactions. The principal and manifest consequence of this symmetry breaking is the generation of 99% of the mass of nucleons, and hence the bulk of all visible matter, out of very light quarks.[1] For example, for the proton, of mass mp= 938 MeV, the bound quarks, with mu ≈ 2 MeV , md ≈ 5 MeV, only contribute about 9 MeV to its mass, the bulk of it arising out of QCD chiral symmetry breaking, instead.[2] Yoichiro Nambu was awarded the 2008 Nobel prize in physics for his understanding of this phenomenon. The origin of the symmetry breaking may be described as an analog to magnetization, the fermion condensate (vacuum condensate of bilinear expressions involving the quarks in the QCD vacuum), formed through nonperturbative action of QCD gluons, with v ≈ −(250 MeV)3. .

Dynamical system The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. History Many people regard Henri Poincaré as the founder of dynamical systems.[3] Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). Basic definitions Flows

Interpretations of quantum mechanics An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters. This question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. They usually consider an interpretation of quantum mechanics as an interpretation of the mathematical formalism of quantum mechanics, specifying the physical meaning of the mathematical entities of the theory. History of interpretations Main quantum mechanics interpreters Nature of interpretation Two qualities vary among interpretations:

Orbit (dynamics) For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and for holomorphic dynamical systems the orbits are Riemann surfaces. Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams) Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function where we define then the set is called orbit through x. for every point x on the orbit. Given a real dynamical system (R, M, Φ), I(x)) is an open interval in the real numbers, that is . is called positive semi-orbit through x and is called negative semi-orbit through x. For discrete time dynamical system : forward orbit of x is a set : backward orbit of x is a set : and orbit of x is a set : where : Usually different notation is used : is written as where is in the above notation. acting on a probability space is a lattice inside

Glueball In particle physics, a glueball is a hypothetical composite particle.[1] It consists solely of gluon particles, without valence quarks. Such a state is possible because gluons carry color charge and experience the strong interaction. Glueballs are extremely difficult to identify in particle accelerators, because they mix with ordinary meson states.[2] Theoretical calculations show that glueballs should exist at energy ranges accessible with current collider technology. However, due to the aforementioned difficulty (among others), they have (as of 2013[update]) so far not been observed and identified with certainty.[3] The prediction that glueballs exist is one of the most important predictions of the Standard Model of Particle Physics that has not yet been confirmed experimentally.[4] Properties of glueballs Constituent Particles and Color Charge Total Angular Momentum Fundamental particles with J=0 or J=2 are easily distinguished from glueballs. Electric Charge

List of chaotic maps List of chaotic maps List of fractals Wick rotation In physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. This transformation is also used to find solutions to problems in quantum mechanics and other areas. Overview Wick rotation is motivated by the observation that the Minkowski metric [with (−1, +1, +1, +1) convention for the metric tensor] and the four-dimensional Euclidean metric are equivalent if one permits the coordinate t to take on imaginary values. , sometimes yields a problem in real Euclidean coordinates x, y, z, which is easier to solve. Statistical and quantum mechanics Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature with imaginary time . . is , where is Boltzmann's constant. is, up to a normalizing constant, under a Hamiltonian . where

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. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set—one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations. This field of study is also called just Dynamical systems, Mathematical Dynamical Systems Theory and Mathematical theory of dynamical systems. Overview Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. History Concepts Dynamical systems Dynamicism

Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. In some contrast to the wave formulation, it produces spectra of energy operators by purely algebraic, ladder operator, methods.[1] Relying on these methods, Pauli derived the hydrogen atom spectrum in 1926,[2] before the development of wave mechanics. Development of matrix mechanics In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. Epiphany at Helgoland In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. "It was about three o' clock at night when the final result of the calculation lay before me. The Three Fundamental Papers After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point:[4] In the paper, Heisenberg formulated quantum theory without sharp electron orbits. W.

Limit set In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. Types In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits. Definition for iterated functions Let be a metric space, and let be a continuous function. -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function . if and only if there is a strictly increasing sequence of natural numbers such that as . where denotes the closure of set . If is a homeomorphism (that is, a bicontinuous bijection), then the Both sets are in R so that and

Mass gap In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. Since exact energy eigenstates are infinitely spread out and are therefore usually excluded from a formal mathematical description, a stronger definition is that the mass gap is the greatest lower bound of the energy of any state which is orthogonal to the vacuum. Mathematical definitions For a given real field with being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. with the constant being finite. Examples from classical theories An example of mass gap arising for massless theories, already at the classical level, can be seen in spontaneous breaking of symmetry or Higgs mechanism. This equation has the exact solution —where and being References

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