classical mechanics

• Frames of reference Main articles: Inertial frame of reference and Galilean transformation 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.

• [20] The work is also the first modern treatise in which a physical problem (the accelerated motion of a falling body) is idealized by a set of parameters then analyzed mathematically
and constitutes one of the seminal works of applied mathematics.

• [clarification needed] Work and energy Main articles: Work (physics), kinetic energy, and potential energy If a constant force F is applied to a particle that makes
a displacement Δr,[note 1] the work done by the force is defined as the scalar product of the force and displacement vectors: More generally, if the force varies as a function of position as the particle moves from r1 to r2 along a path C,
the work done on the particle is given by the line integral If the work done in moving the particle from r1 to r2 is the same no matter what path is taken, the force is said to be conservative.

• The stationary action principle requires that the action functional of the system derived from must remain at a stationary point (a maximum, minimum, or saddle) throughout
the time evolution of the system.

• Kinematics Main article: Kinematics 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.

• However, until now there is no theory of quantum gravity unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.

• Newton, and most of his contemporaries, with the notable exception of Huygens, worked on the assumption that classical mechanics would be able to explain all phenomena, including
light, in the form of geometric optics.

• In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity.

• The motion of a point particle is determined by a small number of parameters: its position, mass, and the forces applied to it.

• 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.

• Since the definition of acceleration is a = dv/dt, the second law can be written in the simplified and more familiar form: So long as the force acting on a particle is known,
Newton’s second law is sufficient to describe the motion of a particle.

• In this formalism, the dynamics of a system are governed by Hamilton’s equations, which express the time derivatives of position and momentum variables in terms of partial
derivatives of a function called the Hamiltonian: The Hamiltonian is the Legendre transform of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.

• With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic
patterns of integrated circuit computer memory.

• 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.

• In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity in Newton’s law of universal gravitation.

• 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.

• 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.

• In addition, Newton’s third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B, it
follows that B must exert an equal and opposite reaction force, −F, on A.

• The resolution of these problems led to the special theory of relativity, often still considered a part of classical mechanics.

• Once independent relations for each force acting on a particle are available, they can be substituted into Newton’s second law to obtain an ordinary differential equation,
which is called the equation of motion.

• Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve,
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
Main article: 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.

• The work–energy theorem states that for a particle of constant mass m, the total work W done on the particle as it moves from position r1 to r2 is equal to the change in kinetic
energy Ek of the particle: Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep: If all the forces acting on a particle are conservative, and Ep is the total potential energy
(which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force The decrease in the potential energy is equal to the increase in the kinetic energy
This result is known as conservation of energy and states that the total energy, is constant in time.

• With objects about the size of an atom’s diameter, it becomes necessary to use quantum mechanics.

• The rocket equation extends the notion of rate of change of an object’s momentum to include the effects of an object “losing mass”.

• Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies.

• Some modern sources include relativistic mechanics in classical physics, as representing the field in its most developed and accurate form.

• [15] Either interpretation has the same mathematical consequences, historically known as “Newton’s Second Law”: The quantity mv is called the (canonical) momentum.

• The development of classical mechanics involved substantial change in the methods and philosophy of physics.

• 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.

• Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space and a smooth function within that space called a Lagrangian.

• As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms and the photo-electric
effect.

• 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.

• Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration.

• Classical mechanics also describes the more complex motions of extended non-pointlike objects.

• Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the speed of light.

• Here they are distinguished from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression.

• While to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical theory and controlled experiment,
as we know it.

• In case that objects become extremely heavy (i.e., their Schwarzschild radius is not negligibly small for a given application), deviations from Newtonian mechanics become
apparent and can be quantified by using the parameterized post-Newtonian formalism.

• Emphasis has shifted to understanding the fundamental forces of nature as in the Standard Model and its more modern extensions into a unified theory of everything.

• Chaos theory shows that the long term predictions of classical mechanics are not reliable.

• This break with ancient thought was happening around the same time that Galileo was proposing abstract mathematical laws for the motion of objects.

• [21] Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics Newton founded his
principles of natural philosophy on three proposed laws of motion: the law of inertia, his second law of acceleration (mentioned above), and the law of action and reaction; and hence laid the foundations for classical mechanics.

• His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics.

• QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction.

• [13] Velocity and speed Main articles: 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: .

• Illustrations of the weak form of Newton’s third law are often found for magnetic forces.

• Classical mechanics assumes that matter and energy have definite, knowable attributes such as location in space and speed.

• Assuming time is measured the same in all reference frames, if we require x = x’ when t = 0, then the relation between the space-time coordinates of the same event observed
from the reference frames S’ and S, which are moving at a relative velocity u in the x direction, is: This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform).

• [11] In contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy.

• 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.

• When treating large degrees of freedom at the macroscopic level, statistical mechanics becomes useful.

• 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.

• As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example: where
λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity.

• [14] 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.

• Since the end of the 20th century, classical mechanics in physics has no longer been an independent theory.

• Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level.

• Statistical mechanics is mainly used in thermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics.

• Description of objects and their motion For simplicity, classical mechanics often models real-world objects as point particles, that is, objects with negligible size.

• The kinetic energy Ek of a particle of mass m travelling at speed v is given by For extended objects composed of many particles, the kinetic energy of the composite body is
the sum of the kinetic energies of the particles.

• Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields.

• When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory (QFT) is of use.

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