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Inertial Frame
An inertial frame of reference, in classical physics, is a frame of reference in which bodies, whose net force acting upon them is zero, are not accelerated; that is they are at rest or they move at a constant velocity in a straight line.[1] In analytical terms, it is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner.[2] Conceptually, in classical physics and special relativity, the physics of a system in an inertial frame have no causes external to the system.[3] An inertial frame of reference may also be called an inertial reference frame, inertial frame, Galilean reference frame, or inertial space.[citation needed] All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration
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Statistical Mechanics
Statistical mechanics
Statistical mechanics
is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system whose exact state is uncertain.[1][2][3][note 1] Statistical mechanics
Statistical mechanics
is commonly used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics. Microscopic mechanical laws do not contain concepts such as temperature, heat, or entropy; however, statistical mechanics shows how these concepts arise from the natural uncertainty about the state of a system when that system is prepared in practice
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Framing (other)
Framing may refer to: Framing (crime), providing false evidence or testimony to prove someone guilty of a crime Framing (construction), the most common carpentry work Framing (social sciences)
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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
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Moment (physics)
In physics, a moment is an expression involving the product of a distance and a physical quantity, and in this way it accounts for how the physical quantity is located or arranged. Moments are usually defined with respect to a fixed reference point; they deal with physical quantities as measured at some distance from that reference point. For example, the moment of force acting on an object, often called torque, is the product of the force and the distance from a reference point. In principle, any physical quantity can be multiplied by distance to produce a moment; commonly used quantities include forces, masses, and electric charge distributions.Contents1 History 2 Elaboration2.1 Examples3 Multipole moments 4 Applications of multipole moments 5 See also 6 References 7 External linksHistory[edit] The concept of moment in physics is derived from the mathematical concept of moments.[1] [clarification needed]
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Momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It can be more generally stated as a measure of how hard it is to stop a moving object. It is a three-dimensional vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is the velocity (also a vector), then the momentum is p = m v , displaystyle mathbf p =mmathbf v , In SI units, it is measured in kilogram meters per second (kg⋅m/s). Newton's second law
Newton's second law
of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum
Momentum
depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change
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Speed
In everyday use and in kinematics, the speed of an object is the magnitude of its velocity (the rate of change of its position); it is thus a scalar quantity.[1] The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval;[2] the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero. Speed
Speed
has the dimensions of distance divided by time. The SI unit
SI unit
of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour
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Torque
Torque, moment, or moment of force is rotational force.[1] Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the position vector (distance vector) and the force vector. The symbol for torque is typically τ displaystyle tau , the lowercase Greek letter tau. When it is called moment of force, it is commonly denoted by M. The magnitude of torque of a rigid body depends on three quantities: the force applied, the lever arm vector[2] connecting the origin to the point of force application, and the angle between the force and lever arm vectors
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Velocity
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion (e.g. 7001600000000000000♠60 km/h to the north). Velocity
Velocity
is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity
Velocity
is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called "speed", being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s) or as the SI base unit of (m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector
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Virtual Work
Virtual work
Virtual work
arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action
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Analytical Mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics
Newtonian mechanics
considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws
Newton's laws
and Euler's laws is vectorial mechanics. By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles.[1] A scalar is a quantity, whereas a vector is represented by quantity and direction
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Routhian Mechanics
In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics
Lagrangian mechanics
and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. The Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and Routhian functions are their variables
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Moment Of Inertia
The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation. It is an extensive (additive) property: For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis)
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Hamilton–Jacobi Equation
In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi. In physics, the Hamilton-Jacobi equation is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion[citation needed], Lagrangian mechanics
Lagrangian mechanics
and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave
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Appell's Equation Of Motion
In classical mechanics, Appell's equation of motion (aka Gibbs-Appell equation of motion) is an alternative general formulation of classical mechanics described by Paul Émile Appell
Paul Émile Appell
in 1900[1] and Josiah Willard Gibbs in 1879[2] Q r = ∂ S
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Udwadia–Kalaba Equation
In theoretical physics, the Udwadia–Kalaba equation [1] is a method for deriving the equations of motion of a constrained mechanical system. This equation was discovered by Firdaus E. Udwadia and Robert E. Kalaba in 1992. The fundamental equation is the simplest and most comprehensive equation so far discovered [2] for writing down the equations of motion of a constrained mechanical system. It makes a convenient distinction between externally applied forces and the internal forces of constraint, similar to the use of constraints in Lagrangian mechanics, but without the use of Lagrange multipliers. The Udwadia–Kalaba equation applies to a wide class of constraints, both holonomic constraints and nonholonomic ones, as long as they are linear with respect to the accelerations
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