Dynamics (mechanics) Generally speaking, researchers involved in dynamics study how a physical system might develop or alter over time and study the causes of those changes. In addition, Newton established the fundamental physical laws which govern dynamics in physics. By studying his system of mechanics, dynamics can be understood. In particular, dynamics is mostly related to Newton's second law of motion. However, all three laws of motion are taken into consideration, because these are interrelated in any given observation or experiment.[1] For classical electromagnetism, it is Maxwell's equations that describe the dynamics. And the dynamics of classical systems involving both mechanics and electromagnetism are described by the combination of Newton's laws, Maxwell's equations, and the Lorentz force. Newton described force as the ability to cause a mass to accelerate.
First law: If there is no net force on an object, then its velocity is constant. Swagatam (25 March 011). Chaos theory. A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions.
In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. Where , and , is: . Acoustics. Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries.
The word "acoustic" is derived from the Greek word ἀκουστικός (akoustikos), meaning "of or for hearing, ready to hear"[2] and that from ἀκουστός (akoustos), "heard, audible",[3] which in turn derives from the verb ἀκούω (akouo), "I hear".[4] The Latin synonym is "sonic", after which the term sonics used to be a synonym for acoustics[5] and later a branch of acoustics.[6] Frequencies above and below the audible range are called "ultrasonic" and "infrasonic", respectively.
History of acoustics[edit] Fluid dynamics. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time. Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.[1] Equations of fluid dynamics[edit] The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum (also known as Newton's Second Law of Motion), and conservation of energy (also known as First Law of Thermodynamics).
Conservation laws[edit] Above, is the viscous dissipation function. Continuum mechanics. Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, but research in the area continues today. Explanation[edit] Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate.
Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. Concept of a continuum[edit] Materials, such as solids, liquids and gases, are composed of molecules separated by "empty" space. Figure 1. Being modeled. Statics. Statics is the branch of mechanics that is concerned with the analysis of loads (force and torque, or "moment") on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity.
When in static equilibrium, the system is either at rest, or its center of mass moves at constant velocity. Vectors[edit] Example of a beam in static equilibrium. The sum of force and moment is zero. A scalar is a quantity, such as mass or temperature, which only has a magnitude. A vector is a quantity that has both a magnitude and a direction. A bold faced character VAn underlined character VA character with an arrow over it . Vectors can be added using the parallelogram law or the triangle law. Force[edit] Force is the action of one body on another. Forces are classified as either contact or body forces. Moment of a force[edit] Moment about a point[edit] Varignon's theorem[edit] Solids[edit] Kinematics. The study of kinematics is often referred to as the geometry of motion.[7] (See analytical dynamics for more detail on usage.) To describe motion, kinematics studies the trajectories of points, lines and other geometric objects and their differential properties such as velocity and acceleration.
Kinematics is used in astrophysics to describe the motion of celestial bodies and systems, and in mechanical engineering, robotics and biomechanics[8] to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the skeleton of the human body. The study of kinematics can be abstracted into purely mathematical functions. For instance, rotation can be represented by elements of the unit circle in the complex plane. Other planar algebras are used to represent the shear mapping of classical motion in absolute time and space and to represent the Lorentz transformations of relativistic space and time.
Kinematics of a particle trajectory[edit] where. Hamiltonian mechanics. Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. Overview[edit] Illustration of a generalized coordinateq for one degree of freedom, of a particle moving in a complicated path. In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates , where each component of the coordinate is indexed to the frame of reference of the system.
The time evolution of the system is uniquely defined by Hamilton's equations:[1] Newton's laws of motion. First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.[2][3]Second law: F = ma. The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object.Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.[4] Newton used them to explain and investigate the motion of many physical objects and systems.[5] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.
Overview Newton's first law Impulse. Lagrangian mechanics. Lagrangian mechanics is a re-formulation of classical mechanics using the principle of stationary action (also called the principle of least action).[1] Lagrangian mechanics applies to systems whether or not they conserve energy or momentum, and it provides conditions under which energy, momentum or both are conserved.[2] It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788. The use of generalized coordinates may considerably simplify a system's analysis. For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics would require solving for the time-varying constraint force required to keep the bead in the groove. For the same problem using Lagrangian mechanics, one looks at the path of the groove and chooses a set of independent generalized coordinates that completely characterize the possible motion of the bead.
Conceptual framework[edit]