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Constraint (classical Mechanics)
In classical mechanics, a constraint on a system is a parameter that the system must obey. For example, a box sliding down a slope must remain on the slope. There are two different types of constraints: holonomic and non-holonomic. Types of constraint *First class constraints and second class constraints *Primary constraints, secondary constraints, tertiary constraints, quaternary constraints. *Holonomic constraints, also called integrable constraints, (depending on time and the coordinates but not on the momenta) and Nonholonomic system *Pfaffian constraint In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form: : \sum_^nA_du_s + A_rdt = 0;\; r = 1,\ldots, L where L is the number of equations in a system of constraints. Holonomic systems can always be written in Pfa ...s * Scleronomic constraints (not depending on time) and rheonomic constraints (depending on time). *Ideal constraints: those for which the work done by the constraint forces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friction Angle
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into ''static friction'' ("stiction") between non-moving surfaces, and ''kinetic friction'' between moving surfaces. With the exception of atomic or molecular friction, dry friction generally arises from the interaction of surface features, known as Asperity (materials science), asperities (see Figure 1). *Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. *Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces. *Skin friction is a component of Drag (physics), drag, the force resisting the motion of a fluid across the surface of a body. *Internal friction is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical System
A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the system. The split between system and environment is the analyst's choice, generally made to simplify the analysis. For example, the water in a lake, the water in half of a lake, or an individual molecule of water in the lake can each be considered a physical system. A Thermostat is one that has negligible interaction with its environment. Often a system in this sense is chosen to correspond to the more usual meaning of heat such as a particular machine. In the study of temperature , the "system" may refer to the microscopic properties of an object (e.g. the mean of a pendulum bob), while the relevant "environment" may be the internal degrees of freedom, described classically by the pendulum's thermavibration See also * Conceptual sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. ''Parameter'' has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. Modelization When a system is modeled by equations, the values that describe the system are called ''parameters''. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holonomic Constraints
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that describe the system. For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation :r^2-a^2=0 where r is the distance from the centre of a sphere of radius a, whereas the second non-holonomic case may be given by :r^2 - a^2 \geq 0 Velocity-dependent constraints (also called semi-holonomic constraints) such as :f(u_1,u_2,\ldots,u_n,\dot_1,\dot_2,\ldots,\dot_n,t)=0 are not usually holonomic. Holonomic system In classical mechanics a system may be defined as holonomic if all constraints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Class Constraint
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated. First and second class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate. The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints. Poisson brackets Consider a Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Class Constraints
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated. First and second class constraints were introduced by as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate. The terminology of first and second class constraints is confusingly similar to that of primary and secondary constraints, reflecting the manner in which these are generated. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints. Poisson brackets Consider a Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Constraint
In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time, the tertiary constraints arise from the condition that the secondary ones are also preserved by time, and so on. Primary and secondary constraints were introduced by Anderson and Bergmann Bergmann is a German or Swedish surname. It means "mountain man" in both languages, as well as "miner" in German. '' Bergman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secondary Constraint
In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time, the tertiary constraints arise from the condition that the secondary ones are also preserved by time, and so on. Primary and secondary constraints were introduced by Anderson and Bergmann Bergmann is a German or Swedish surname. It means "mountain man" in both languages, as well as "miner" in German. '' Bergman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tertiary Constraint
In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfied The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time, the tertiary constraints arise from the condition that the secondary ones are also preserved by time, and so on. Primary and secondary constraints were introduced by Anderson and Bergmann Bergmann is a German or Swedish surname. It means "mountain man" in both languages, as well as "miner" in German. '' Bergman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternary Constraint
The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million years ago to the present. The Quaternary Period is divided into two epochs: the Pleistocene (2.58 million years ago to 11.7 thousand years ago) and the Holocene (11.7 thousand years ago to today, although a third epoch, the Anthropocene, has been proposed but is not yet officially recognised by the ICS). The Quaternary Period is typically defined by the cyclic growth and decay of continental ice sheets related to the Milankovitch cycles and the associated climate and environmental changes that they caused. Research history In 1759 Giovanni Arduino proposed that the geological strata of northern Italy could be divided into four successive formations or "orders" ( it, quattro ordini). The term "quaternary" was introduced by Jules Desnoyer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonholonomic System
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, such that when the system evolves along a path in its parameter space (the parameters varying continuously in values) but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state. Nonholonomic mechanics is autonomous division of Newtonian mechanics. Details More precisely, a nonholonomic system, also called an ''anholonomic'' system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
