Classical mechanics

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.[3] In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.[4]

The velocity, or the rate of change of displacement with time, is defined as the derivative of the position with respect to time:

${\displaystyle \mathbf {v} ={\mathrm {d} \mathbf {r} \over \mathrm {d} t}\,\!}$.

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:

${\displaystyle \mathbf {u} '=\mathbf {u} -\mathbf {v} \,.}$

Similarly, the first object sees the velocity of the second object as:

${\displaystyle \mathbf {v'} =\mathbf {v} -\mathbf {u} \,.}$

When both objects are moving in the same direction, this equation can be simplified to:

${\displaystyle \mathbf {u} '=(u-v)\mathbf {d} \,.}$

Or, by ignoring direction, the difference can be given in terms of speed only:

${\displaystyle u'=u-v\,.}$

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):

${\displaystyle \mathbf {a} ={\mathrm {d} \mathbf {v} \over \mathrm {d} t}={\mathrm {d^{2}} \mathbf {r} \over \mathrm {d} t^{2}}.}$

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.

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, ${\displaystyle F=ma}$, is valid.[5]^: 185

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.[5] 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.

Consider two reference frames S and S'. For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frame S and (x',y',z',t') in frame S'. 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:

${\displaystyle {\begin{aligned}x'&=x-tu,\\y'&=y,\\z'&=z,\\t'&=t.\end{aligned}}}$

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This group is a limiting case of the Poincaré group used in special relativity. The limiting case applies when the velocity u is very small compared to c, the speed of light.

The transformations have the following consequences:

• 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.

A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a field, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature.[6] Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":

${\displaystyle \mathbf {F} ={\mathrm {d} \mathbf {p} \over \mathrm {d} t}={\mathrm {d} (m\mathbf {v} ) \over \mathrm {d} t}.}$

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:

${\displaystyle \mathbf {F} =m\mathbf {a} \,.}$

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:

${\displaystyle \mathbf {F} _{\rm {R}}=-\lambda \mathbf {v} \,,}$

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

${\displaystyle -\lambda \mathbf {v} =m\mathbf {a} =m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}\,.}$

This can be integrated to obtain

${\displaystyle \mathbf {v} =\mathbf {v} _{0}e^{{-\lambda t}/{m}}}$

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, 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 strong form of Newton's third law requires that F and −F act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.

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:

${\displaystyle W=\mathbf {F} \cdot \Delta \mathbf {r} \,.}$

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

${\displaystyle W=\int _{C}\mathbf {F} (\mathbf {r} )\cdot \mathrm {d} \mathbf {r} \,.}$

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. Gravity is a conservative force, as is the force due to an idealized spring, as given by Hooke's law. The force due to friction is non-conservative.

The kinetic energy E_k of a particle of mass m travelling at speed v is given by

${\displaystyle E_{\mathrm {k} }={\tfrac {1}{2}}mv^{2}\,.}$

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 E_k of the particle:

${\displaystyle W=\Delta E_{\mathrm {k} }=E_{\mathrm {k_{2}} }-E_{\mathrm {k_{1}} }={\tfrac {1}{2}}m\left(v_{2}^{\,2}-v_{1}^{\,2}\right).}$

Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted E_p:

${\displaystyle \mathbf {F} =-\mathbf {\nabla } E_{\mathrm {p} }\,.}$

If all the forces acting on a particle are conservative, and E_p 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

${\displaystyle \mathbf {F} \cdot \Delta \mathbf {r} =-\mathbf {\nabla } E_{\mathrm {p} }\cdot \Delta \mathbf {r} =-\Delta E_{\mathrm {p} }\,.}$

The decrease in the potential energy is equal to the increase in the kinetic energy

${\displaystyle -\Delta E_{\mathrm {p} }=\Delta E_{\mathrm {k} }\Rightarrow \Delta (E_{\mathrm {k} }+E_{\mathrm {p} })=0\,.}$

This result is known as conservation of energy and states that the total energy,

${\displaystyle \sum E=E_{\mathrm {k} }+E_{\mathrm {p} }\,,}$

is constant in time. It is often useful, because many commonly encountered forces are conservative.

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

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates. These are basically mathematical rewritings of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. Also, analogy with quantum mechanics is more explicit in Hamiltonian formalism.

The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c², where c is the speed of light in free space.

In special relativity, the momentum of a particle is given by

${\displaystyle \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,,}$

where m is the particle's rest mass, v its velocity, v is the modulus of v, and c is the speed of light.

If v is very small compared to c, v²/c² is approximately zero, and so

${\displaystyle \mathbf {p} \approx m\mathbf {v} \,.}$

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

${\displaystyle f=f_{\mathrm {c} }{\frac {m_{0}}{m_{0}+{\frac {T}{c^{2}}}}}\,,}$

where f_c 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 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

${\displaystyle \lambda ={\frac {h}{p}}}$

where h is Planck's constant and p is the momentum.

Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a single diffraction side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. 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.

More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits.

Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.