In physics, classical mechanics is a theory that describes the effect of forces on the motion of macroscopic objects and bulk matter, without considering quantum effects, and often without incorporating relativistic effects either.
It is used in describing the motion of objects such as projectiles and particles, parts of machinery, spacecraft, planets, stars, galaxies, deformable solids, fluids, macromolecules and other objects. The development of classical mechanics involved substantial change in the methods and philosophy of physics.. The qualifier classical distinguishes this type of mechanics from new methods developed after the revolutions in physics of the early 20th century which revealed limitations in classical mechanics. Some modern sources include relativistic mechanics in classical mechanics, as representing the subject matter in its most developed and accurate form.
The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton, and the mathematical methods invented by Newton, Gottfried Wilhelm Leibniz, Leonhard Euler and others to describe the motion of bodies under the influence of forces. Later, methods based on energy were developed by Euler, Joseph-Louis Lagrange, William Rowan Hamilton and others, leading to the development of analytical mechanics (which includes Lagrangian mechanics and Hamiltonian mechanics). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics.
If the present state of an object that obeys the laws of classical mechanics is known, it is possible to determine how it will move in the future, and how it has moved in the past. Chaos theory shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the speed of light. With objects about the size of an atom's diameter, it becomes necessary to use quantum mechanics. To describe velocities approaching the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable.
Branches Traditional division Classical mechanics was traditionally divided into three main branches. Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment. Kinematics describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics.. Dynamics goes beyond merely describing objects' behavior and also considers the forces which explain it. Some authors (for example Greenwood (1997)) include special relativity within classical dynamics.
Forces vs. energy Another division is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on Newton's laws of motion. Newtonian mechanics in this sense emphasizes force as a vector quantity.
In contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Two dominant branches of analytical mechanics are Lagrangian mechanics, which uses generalized coordinates and corresponding generalized velocities in tangent bundle space (the tangent bundle of the configuration space and sometimes called "state space"), and Hamiltonian mechanics, which uses coordinates and corresponding momenta in phase space (the cotangent bundle of the configuration space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries.
By region of application Alternatively, a division can be made by region of application: Celestial mechanics, relating to stars, planets and other celestial bodies Continuum mechanics, for materials modelled as a continuum, e.g., solids and fluids (i.e., liquids and gases). Relativistic mechanics (i.e. including the special and general theories of relativity), for bodies whose speed is greater than a small fraction of the speed of light. Statistical mechanics, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.
Description of objects and their motion
Classical mechanics models real-world objects as point particles, that is, objects with negligible size. Classical mechanics describes the motions of extended non-pointlike objects by considering these objects to be aggregates of rigidly connected particles.
In the Newtonian form of classical mechanics the motion of a point particle is determined by a small number of parameters: its position, mass, and the forces applied to it. Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass". (These generalizations/extensions are derived from Newton's laws, say, by decomposing a solid body into a collection of points.)
In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The behavior of very small particles, such as the electron, is more accurately described by quantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g., a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles. The center of mass of a composite object behaves like a point particle.
Classical mechanics assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance).
Kinematics
+ The SI derived "mechanical"(that is, not electromagnetic or thermal)units with kg, m and spositionmangular position/angleunitless (radian)velocitym·s−1angular velocitys−1accelerationm·s−2angular accelerations−2jerkm·s−3"angular jerk"s−3specific energym2·s−2absorbed dose ratem2·s−3moment of inertiakg·m2momentumkg·m·s−1angular momentumkg·m2·s−1forcekg·m·s−2torquekg·m2·s−2energykg·m2·s−2powerkg·m2·s−3pressure and energy densitykg·m−1·s−2surface tensionkg·s−2spring constantkg·s−2irradiance and energy fluxkg·s−3kinematic viscositym2·s−1dynamic viscositykg·m−1·s−1density (mass density)kg·m−3specific weight (weight density)kg·m−2·s−2number densitym−3actionkg·m2·s−1 The position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O. In cases where P is moving relative to O, r is defined as a function of t, time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval that is observed to elapse between any given pair of events is the same for all observers. Extract of page 30 In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.
Velocity and speed
The velocity, or the rate of change of displacement with time, is defined as the derivative of the position with respect to time: .
In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at . However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as −10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.
Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector and the velocity of the second object by the vector , where u is the speed of the first object, v is the speed of the second object, and d and e are unit vectors in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is:
Similarly, the first object sees the velocity of the second object as:
When both objects are moving in the same direction, this equation can be simplified to:
Or, by ignoring direction, the difference can be given in terms of speed only:
Acceleration
The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time):
Acceleration represents the velocity's change over time. Velocity can change in magnitude, direction, or both. Occasionally, a decrease in the magnitude of velocity "v" is referred to as deceleration, but generally any change in the velocity over time, including deceleration, is referred to as acceleration.
Frames of reference
While the position, velocity and acceleration of a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. An inertial frame is an idealized frame of reference within which an object with zero net force acting upon it moves with a constant velocity; that is, it is either at rest or moving uniformly in a straight line. In an inertial frame Newton's law of motion, , is valid.
Non-inertial reference frames accelerate in relation to another inertial frame. A body rotating with respect to an inertial frame is not an inertial frame. When viewed from an inertial frame, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as fictitious forces, inertia forces, or pseudo-forces.