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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[edit] 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[edit] Simple example[edit] The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum. Importance[edit] does not depend on . . . Related:  SWARM BEHAVIOUR

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[edit] 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[edit] If the initial vector is aligned with a right eigenvector If ) of the matrix .

Euler equations (fluid dynamics) In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and without heat conduction terms. Historically, only the continuity and momentum equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – together as "the Euler equations".[1] The Euler equations can be applied to compressible as well as to incompressible flow – using either an appropriate equation of state or assuming that the divergence of the flow velocity field is zero, respectively. During the second half of the 19th century, it was found that the equation related to the conservation of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. where and or

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[edit] 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[edit] 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[edit] Stable limit cycles are examples of attractors. Finding limit cycles[edit] . Open problems[edit] See also[edit]

Swarm behaviour A flock of auklets exhibit swarm behaviour From a more abstract point of view, swarm behaviour is the collective motion of a large number of self-propelled entities.[1] From the perspective of the mathematical modeller, it is an emergent behaviour arising from simple rules that are followed by individuals and does not involve any central coordination. Swarm behaviour was first simulated on a computer in 1986 with the simulation program boids.[2] This program simulates simple agents (boids) that are allowed to move according to a set of basic rules. The model was originally designed to mimic the flocking behaviour of birds, but it can be applied also to schooling fish and other swarming entities. Models[edit] In recent decades, scientists have turned to modeling swarm behaviour to gain a deeper understanding of the behaviour. Mathematical models[edit] Early studies of swarm behaviour employed mathematical models to simulate and understand the behaviour. Evolutionary models[edit] Agents[edit]

Dynamical system The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview[edit] 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[edit] 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[edit] Flows[edit]

Agent-based model An agent-based model (ABM) is one of a class of computational models for simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) with a view to assessing their effects on the system as a whole. It combines elements of game theory, complex systems, emergence, computational sociology, multi-agent systems, and evolutionary programming. Monte Carlo Methods are used to introduce randomness. Agent-based models are a kind of microscale model [3] that simulate the simultaneous operations and interactions of multiple agents in an attempt to re-create and predict the appearance of complex phenomena. Most agent-based models are composed of: (1) numerous agents specified at various scales (typically referred to as agent-granularity); (2) decision-making heuristics; (3) learning rules or adaptive processes; (4) an interaction topology; and (5) a non-agent environment. History[edit] Early developments[edit] Theory[edit]

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

Swarm robotics Swarm of open-source Jasmine micro-robots recharging themselves Swarm robotics is a new approach to the coordination of multirobot systems which consist of large numbers of mostly simple physical robots. It is supposed that a desired collective behavior emerges from the interactions between the robots and interactions of robots with the environment. Definition[edit] The research of swarm robotics is to study the design of robots, their physical body and their controlling behaviors. Unlike distributed robotic systems in general, swarm robotics emphasizes a large number of robots, and promotes scalability, for instance by using only local communication. Video tracking is an essential tool for systematically studying swarm-behavior, even though other tracking methods are available. Goals and applications[edit] Both miniaturization and cost are key-factors in swarm robotics. See also[edit] External links[edit]

List of chaotic maps List of chaotic maps[edit] List of fractals[edit] Swarmanoid project 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[edit] Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. History[edit] Concepts[edit] Dynamical systems[edit] Dynamicism[edit]

Swarmrobot | Open-source micro-robotic project 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[edit] 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[edit] 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

Airborne robot swarms are making complex moves (w/ video) ( -- The GRASP Lab at the University of Pennsylvania this week released a video that shows their new look in GRASP Lab robotic flying devices. They are now showing flying devices with more complex behavior than before, in a fleet of flying devices that move in packs, navigate spaces with obstacles, flip over and retain position, and carry out formation flying, The researchers have cut down these robotic creature-like drones to small size to what they call “nano-quadrotors.” The video shows them in action: not just engaged in formation flying, but also creating an impressive looking figure-eight pattern. The video says as much about the GRASP Lab as the flying machines, in that the GRASP Labs seems intent on raising the bar on what robot swarms can achieve. Still, the video is clear proof that the team developers, Alex Kushleyev, Daniel Mellinger, and Vijay Kumar, are able to showcase complex autonomous swarm behavior. The key word is agile. More information: