# Classical mechanics

In physics, classical mechanics or Newtonian mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies. The other sub-field is quantum mechanics. Roughly speaking, classical mechanics was developed in the 400 years since the groundbreaking works of Brahe, Kepler, and Galileo, while quantum mechanics developed within the last 100 years, starting with similarly decisive discoveries by Planck, Einstein, and Bohr.

The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is characterized by the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. This is further described in the following sections. More abstract, and general methods include Lagrangian mechanics and Hamiltonian mechanics.

Classical mechanics produces very accurate results within the domain of everyday experience. It is enhanced by special relativity for objects moving with high velocity, more than about a third the speed of light. Classical mechanics is used to describe the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies, and even microscopic objects such as large molecules. Besides this, many specialties exist, dealing with gases, liquids, and solids, and so on. It is one of the largest subjects in science and technology.

## Description of the theory

The following introduces the basic concepts of classical mechanics. For simplicity, it uses 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 which classical mechanics can describe always have a non-zero size. True point particles, such as the electron, are normally better described by quantum mechanics. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, 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 up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.

### Position and its derivatives

The position of a point particle is defined with respect to an arbitrary fixed point in space, which is sometimes called the origin, O. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary, so r is a function of t, the time elapsed since an arbitrary initial time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames.

#### Velocity

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

$\displaystyle \mathbf{v} = {d\mathbf{r} \over dt}$ .

In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, from the perspective of the car it passes it is traveling East at 60−50 = 10 km/h. From the perspective of the faster car, the slower car is moving 10 km/h to the West. What if the car is traveling north? 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 v = vd and the velocity of the second object by the vector u = ue where v is the speed of the first object, u is the speed of the second object, and d and e are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:

v' = v - u

Similarly:

u' = u - v

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

v' = ( v - u ) d

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

v' = v - u

#### Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time or

$\displaystyle \mathbf{a} = {d\mathbf{v} \over dt}$ .

The acceleration vector can be changed by changing its magnitude, changing its direction, or both. If the magnitude of v decreases, this is sometimes referred to as deceleration or retardation; but generally any change in the velocity, including deceleration, is simply referred to as acceleration.

#### Frames of reference

The following consequences can be derived about the perspective of an event in two reference frames, S and S, where S is traveling at a relative velocity of u to S.

• 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 remains the same regardless of reference frame)
• F' = F (since F = ma) (the force on a particle remains the same regardless of reference frame; see Newton's law)
• the speed of light is not a constant
• the form of Maxwell's equations is not preserved across reference frames

### Forces; Newton's second law

Newton's second law relates the mass and velocity of a particle to a vector quantity known as the force. If m is the mass of a particle and F is the vector sum of all applied forces (i.e. the net applied force), Newton's second law states that

$\displaystyle \mathbf{F} = {d(m \mathbf{v}) \over dt}$ .

The quantity mv is called the momentum. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form

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

where $\displaystyle \mathbf a = \frac {d \mathbf v} {dt}$ is the acceleration. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.

Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for F, obtained by considering the particular physical entities with which the particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example:

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

with λ a positive constant. 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. Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is

$\displaystyle - \lambda \mathbf{v} = m \mathbf{a} = m {d\mathbf{v} \over dt}$ .

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. 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.

### Energy

If a force F is applied to a particle that achieves a displacement Δs, the work done by the force is the scalar quantity

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

If the mass of the particle is constant, and ΔWtotal is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:

$\displaystyle \Delta W_{\rm total} = \Delta E_k \,\!$ ,

where Ek is called the kinetic energy. For a point particle, it is defined as

$\displaystyle E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2$ .

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

A particular class of forces, known as conservative forces, can be expressed as the gradient of a scalar function, known as the potential energy and denoted Ep:

$\displaystyle \mathbf{F} = - \nabla E_p$ .

If all the forces acting on a particle are conservative, and Ep is the total potential energy, obtained by summing the potential energies corresponding to each force

 $\displaystyle \mathbf{F} \cdot \Delta \mathbf{s} = - \nabla E_p \cdot \Delta \mathbf{s} = - \Delta E_p \Rightarrow - \Delta E_p = \Delta E_k \Rightarrow \Delta (E_k + E_p) = 0 \,\!$ .

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

$\displaystyle \sum E = E_k + E_p \,\!$

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

### Beyond Newton's Laws

Classical mechanics also includes descriptions of the complex motions of extended non-pointlike objects. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion.

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are often more useful for solving problems. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.

### Classical transformations

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, and 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 of u in the x direction is:

x' = x - ut
y' = y
z' = z
t' = t

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This type of transformation is a limiting case of Special Relativity when the velocity u is very small compared to c, the speed of light.

## History

Main article: History of classical mechanics

The Greeks, and Aristotle in particular, were the first to propose that there are abstract principles governing nature.

One of the first scientists who suggested abstract laws was Galileo Galilei who may have performed the famous experiment of dropping two cannon balls from the tower of Pisa. (The theory and the practice showed that they both hit the ground at the same time.) Though the reality of this experiment is disputed, he did carry out quantitative experiments by rolling balls on an inclined plane; his correct theory of accelerated motion was apparently derived from the results of the experiments.

Sir Isaac Newton was the first to propose the three laws of motion (the law of inertia, his second law mentioned above, and the law of action and reaction), and to prove that these laws govern both everyday objects and celestial objects.

Newton and most of his contemporaries, with the notable exception of Christiaan Huygens hoped that classical mechanics would be able to explain all entities, including (in the form of geometric optics) light. When he discovered Newton's rings, Newton's own explanation avoided wave principles and resembled more the explanation for the decay of the neutral Kaons, K0 and K0 bar. That is, he supposed that the light particles were altered or excited by the glass and resonated.

Newton also developed the calculus which is necessary to perform the mathematical calculations involved in classical mechanics. However it was Gottfried Leibniz who developed the notation of the derivative and integral which are used to this day.

After Newton the field became more mathematical and more abstract.

Although classical mechanics is largely compatible with other "classical physics" theories such as classical electrodynamics and thermodynamics, some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox in which entropy is not a well-defined quantity. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms. The effort at resolving these problems led to the development of quantum mechanics. Similarly, the different behaviour of classical electromagnetism and classical mechanics under velocity transformations led to the theory of relativity.

By the end of the 20th century, the place of classical mechanics in physics is no longer that of an independent theory. Along with classical electromagnetism, it has become imbedded in relativistic quantum mechanics or quantum field theory[1]. It is the non-relativistic, non-quantum mechanical limit for massive particles.

## Limits of validity

### The classical approximation to special relativity

Non-relativistic classical mechanics approximates the relativistic momentum $\displaystyle \frac{m_0 v}{ \sqrt{1-v^2/c^2}}$ with $\displaystyle m_0 v$ , so it is only valid when the velocity is much less than the speed of light. For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by $\displaystyle f=f_c\frac{m_0}{m_0+T/c^2}$ , where $\displaystyle f_c$ is the classical frequency of an electron (or other charged particle) with kinetic energy $\displaystyle T$ and (rest) mass $\displaystyle m_0$ 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

The ray approximation of classical mechanics breaks down when the de Broglie wave length is not much smaller than other dimensions of the system. For non-relativistic particles, this wave length is

$\displaystyle \lambda=\frac{2\pi\hbar}{p}$

where $\displaystyle \hbar$ is Plank's constant divided by $\displaystyle 2\pi$ and $\displaystyle p$ 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 volts, had a wave length 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 transister 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 wave lengths than massless particles, such as light, with the same kinetic energies.