Contents 1 Description of the theory 1.1 Position and its derivatives 1.1.1
1.2 Forces; Newton's second law 1.3 Work and energy 1.4 Beyond Newton's laws 2 Limits of validity 2.1 The Newtonian approximation to special relativity 2.2 The classical approximation to quantum mechanics 3 History 4 Branches 5 See also 6 Notes 7 References 8 Further reading 9 External links Description of the theory[edit] The analysis of projectile motion is a part of classical mechanics. The following introduces the basic concepts of classical mechanics.
For simplicity, it often models real-world objects as point particles
(objects with negligible size). The motion of a point particle is
characterized by a small number of parameters: its position, mass, and
the forces applied to it. Each of these parameters is discussed in
turn.
In reality, the kind of objects that classical mechanics can describe
always have a non-zero size. (The physics 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.
Position and its derivatives[edit] Main article: Kinematics The SI derived "mechanical" (that is, not electromagnetic or thermal) units with kg, m and s position m angular position/angle unitless (radian) velocity m·s−1 angular velocity s−1 acceleration m·s−2 angular acceleration s−2 jerk m·s−3 "angular jerk" s−3 specific energy m2·s−2 absorbed dose rate m2·s−3 moment of inertia kg·m2 momentum kg·m·s−1 angular momentum kg·m2·s−1 force kg·m·s−2 torque kg·m2·s−2 energy kg·m2·s−2 power kg·m2·s−3 pressure and energy density kg·m−1·s−2 surface tension kg·s−2 spring constant kg·s−2 irradiance and energy flux kg·s−3 kinematic viscosity m2·s−1 dynamic viscosity kg·m−1·s−1 density (mass density) kg·m−3 density (weight density) kg·m−2·s−2 number density m−3 action kg·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.[1] In addition to relying on absolute time, classical
mechanics assumes
v = d r d t displaystyle mathbf v = mathrm d mathbf r over mathrm d t ,! . 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 60 − 50 = 10 km/h. 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 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 u ′ = u − v . displaystyle mathbf u '=mathbf u -mathbf v ,. Similarly, the first object sees the velocity of the second object as v ′ = v − u . displaystyle mathbf v' =mathbf v -mathbf u ,. When both objects are moving in the same direction, this equation can be simplified to u ′ = ( u − v ) d . displaystyle mathbf u '=(u-v)mathbf d ,. Or, by ignoring direction, the difference can be given in terms of speed only: u ′ = u − v . displaystyle u'=u-v,. Acceleration[edit] 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): a = d v d t = d 2 r d t 2 . displaystyle mathbf a = mathrm d mathbf v over mathrm d t = mathrm d^ 2 mathbf r over mathrm d t^ 2 .
x ′ = x − u t displaystyle x'=x-ut, y ′ = y displaystyle y'=y, z ′ = z displaystyle z'=z, t ′ = t . displaystyle t'=t,. This set of formulas defines a group transformation known as the
v′ = v − u (the velocity v′ of a particle from the perspective of S′ is slower by u than its velocity v from the perspective of S) a′ = a (the acceleration of a particle is the same in any inertial reference frame) F′ = F (the force on a particle is the same in any inertial reference frame) the speed of light is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics. For some problems, it is convenient to use rotating coordinates
(reference frames). Thereby one can either keep a mapping to a
convenient inertial frame, or introduce additionally a fictitious
centrifugal force and Coriolis force.
Forces; Newton's second law[edit]
Main articles:
F = d p d t = d ( m v ) d t . displaystyle mathbf F = mathrm d mathbf p over mathrm d t = mathrm d (mmathbf v ) over mathrm d t . The quantity mv is called the (canonical) momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is a = dv/dt, the second law can be written in the simplified and more familiar form: F = m a . displaystyle mathbf F =mmathbf a ,. So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. 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. 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: F R = − λ v , displaystyle mathbf F _ rm R =-lambda mathbf v ,, where λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is − λ v = m a = m d v d t . displaystyle -lambda mathbf v =mmathbf a =m mathrm d mathbf v over mathrm d t ,. This can be integrated to obtain v = v 0 e − λ t m displaystyle mathbf v =mathbf v _ 0 e^ frac -lambda t m where v0 is the initial velocity. This means that the velocity of this
particle decays exponentially to zero as time progresses. In this
case, an equivalent viewpoint is that the kinetic energy of the
particle is absorbed by friction (which converts it to heat energy in
accordance with the conservation of energy), and the particle is
slowing down. This expression can be further integrated to obtain the
position r of the particle as a function of time.
Important forces include the gravitational force and the Lorentz force
for electromagnetism. In addition,
W = F ⋅ Δ r . displaystyle W=mathbf F cdot Delta mathbf r ,. 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 W = ∫ C F ( r ) ⋅ d r . displaystyle W=int _ C mathbf F (mathbf r )cdot mathrm d mathbf r ,. 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.
E k = 1 2 m v 2 . displaystyle E_ mathrm k = tfrac 1 2 mv^ 2 ,. For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles. 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: W = Δ E k = E k , 2 − E k , 1 = 1 2 m ( v 2 2 − v 1 2 ) . displaystyle W=Delta E_ mathrm k =E_ mathrm k,2 -E_ mathrm k,1 = tfrac 1 2 mleft(v_ 2 ^ ,2 -v_ 1 ^ ,2 right),. Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep: F = − ∇ E p . displaystyle mathbf F =-mathbf nabla E_ mathrm p ,. 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 F ⋅ Δ r = − ∇ E p ⋅ Δ r = − Δ E p . displaystyle mathbf F cdot Delta mathbf r =-mathbf nabla E_ mathrm p cdot Delta mathbf r =-Delta E_ mathrm p ,. The decrease in the potential energy is equal to the increase in the kinetic energy − Δ E p = Δ E k ⇒ Δ ( E k + E p ) = 0 . displaystyle -Delta E_ mathrm p =Delta E_ mathrm k Rightarrow Delta (E_ mathrm k +E_ mathrm p )=0,. This result is known as conservation of energy and states that the total energy, ∑ E = E k + E p , displaystyle sum E=E_ mathrm k +E_ mathrm p ,, is constant in time. It is often useful, because many commonly
encountered forces are conservative.
Beyond Newton's laws[edit]
Domain of validity for Classical Mechanics Many branches of classical mechanics are simplifications or
approximations of more accurate forms; two of the most accurate being
general relativity and relativistic statistical mechanics. Geometric
optics is an approximation to the quantum theory of light, and does
not have a superior "classical" form.
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. 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. When
treating large degrees of freedom at the macroscopic level,
statistical mechanics becomes useful.
p = m v 1 − v 2 / c 2 , displaystyle mathbf p = frac mmathbf v sqrt 1-v^ 2 /c^ 2 ,, where m is the particle's rest mass, v its velocity, and c is the speed of light. If v is very small compared to c, v2/c2 is approximately zero, and so p ≈ m v . displaystyle mathbf p approx mmathbf v ,. Thus the Newtonian equation p = mv is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light. For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by f = f c m 0 m 0 + T / c 2 , displaystyle f=f_ mathrm c frac m_ 0 m_ 0 +T/c^ 2 ,, where fc is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m0 circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage. The classical approximation to quantum mechanics[edit] The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is λ = h p displaystyle lambda = frac h p where h is
Three stage
The first published causal explanation of the motions of planets was Johannes Kepler's Astronomia nova, published in 1609. He concluded, based on Tycho Brahe's observations on the orbit of Mars, that the planet's orbits were ellipses. This break with ancient thought was happening around the same time that Galileo was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannonballs of different weights from the tower of Pisa, showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an inclined plane. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. Sir
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. Both Newton's second and
third laws were given the proper scientific and mathematical treatment
in Newton's Philosophiæ Naturalis Principia Mathematica. Here they
are distinguished from earlier attempts at explaining similar
phenomena, which were either incomplete, incorrect, or given little
accurate mathematical expression. Newton also enunciated the
principles of conservation of momentum and angular momentum. 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. The combination of
Hamilton's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. 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. Even when discovering the so-called Newton's rings
(a wave interference phenomenon) he maintained his own corpuscular
theory of light.
After Newton, classical mechanics became a principal field of study in
mathematics as well as physics. Several re-formulations progressively
allowed finding solutions to a far greater number of problems. The
first notable re-formulation was in 1788 by Joseph Louis Lagrange.
Statics, the study of equilibrium and its relation to forces Dynamics, the study of motion and its relation to forces Kinematics, dealing with the implications of observed motions without regard for circumstances causing them Another division is based on the choice of mathematical formalism: Newtonian mechanics Lagrangian mechanics Hamiltonian mechanics 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).
See also[edit]
Dynamical systems
History of classical mechanics
List of equations in classical mechanics
List of publications in classical mechanics
Molecular dynamics
Newton's laws of motion
Notes[edit] ^ The notion of "classical" may be somewhat confusing, since this term
usually refers to the era of classical antiquity in European history.
While many discoveries within the mathematics of that period are
applicable today and of great use, much of the science that emerged
from that time has since been superseded by more accurate models. This
in no way detracts from the science of that time as most of modern
physics is built directly upon those developments. The emergence of
classical mechanics was a decisive stage in the development of
science, in the modern sense of the term. Above all, it is
characterized by an insistence that more rigor be used to describe the
behavior of bodies. Such an exacting foundation is only available
through mathematical treatment and reliance on experiment, rather than
speculation.
References[edit] ^ Knudsen, Jens M.; Hjorth, Poul (2012). Elements of Newtonian
Further reading[edit] Alonso, M.; Finn, J. (1992). Fundamental University Physics.
Addison-Wesley.
Feynman, Richard (1999). The Feynman Lectures on Physics. Perseus
Publishing. ISBN 0-7382-0092-1.
Feynman, Richard; Phillips, Richard (1998). Six Easy Pieces. Perseus
Publishing. ISBN 0-201-32841-0.
Goldstein, Herbert; Charles P. Poole; John L. Safko (2002). Classical
External links[edit] Wikimedia Commons has media related to Classical mechanics. Wikiquote has quotations related to: Classical mechanics Crowell, Benjamin.
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