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Classical mechanics

Classical mechanics is the physics of forces, acting upon bodies. It is often referred to as "Newtonian mechanics" after Newton and his laws of motion. Classical mechanics is subdivided into statics (which deals with objects in equilibrium) and dynamics (which deals with objects in motion).

Classical mechanics produces very accurate results within the domain of everyday experience. It is superseded by relativistic mechanics for systems moving at large velocities near the speed of light, quantum mechanics for systems at small distance scales, and relativistic quantum field theory for systems with both properties. Nevertheless, classical mechanics is still very useful, because (i) it is much simpler and easier to apply than these other theories, and (ii) it has a very large range of approximate validity. Classical mechanics can be used to describe the motion of human-sized objects (such as tops[?] and baseballs), many astronomical objects (such as planets and galaxies), and even certain microscopic objects (such as organic molecules.)

Although classical mechanics is roughly compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are inconsistencies that were discovered in the late 19th century that can only be resolved by more modern physics. In particular, classical electrodynamics predicts that the speed of light is constant to all observers, a prediction that is difficult to reconcile with classical mechanics and which led to the development of special relativity. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox[?] in which entropy is not a well-defined quantity and to the ultraviolet catastrophe in which a blackbody is predicted to emit infinite amounts of energy. The effort at resolving these problems led to the development of quantum mechanics.

Table of contents

Description of the theory

We will now introduce the basic concepts of classical mechanics. For simplicity, we only deal with a point particle, which is an object with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied on it. We will discuss each of these parameters 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 properly described by quantum mechanics. Objects with non-zero size have more complicated behavior than our hypothetical point particles, because their internal configuration can change - for example, a baseball can spin while it is moving. However, we will be able to use our results for point particles to study such objects by treating them as composite objects, made up of a large number of interacting point particles. We can then show that such composite objects behave like point particles, provided they are small compared to the distance scales of the problem, which indicates that our use of point particles is self-consistent.

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. The velocity, or the rate of change of position with time, is defined as

<math>\mathbf{v} = {d\mathbf{r} \over dt}</math>.

The acceleration, or rate of change of velocity, is

<math>\mathbf{a} = {d\mathbf{v} \over dt}</math>.

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; but generally any change in the velocity, including deceleration, is simply referred to as acceleration.

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. Suppose m is the mass of a particle and F is the vector sum of all applied forces (i.e. the net applied force.) Then Newton's second law states that

<math>\mathbf{F} = {d(m \mathbf{v}) \over dt}</math>.

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

<math>\mathbf{F} = m \mathbf{a}</math>

where a is the acceleration, as defined above. 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, we require a description of F, which is to be obtained by considering the particular physical entities with which our particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, say

<math>\mathbf{F}_{\rm R} = - \lambda \mathbf{v}</math>

with λ a positive constant. Once we have independent relations for each force acting on a particle, we can substitute it into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Continuing our example, suppose that friction is the only force acting on the particle. Then the equation of motion is

<math>- \lambda \mathbf{v} = m \mathbf{a} = m {d\mathbf{v} \over dt}</math>.

This can be integrated to obtain

<math>\mathbf{v} = \mathbf{v}_0 e^{- \lambda t / m}</math>

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 we know 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.

Energy

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

<math>\delta W = \mathbf{F} \cdot \delta \mathbf{r}</math>.

Suppose the mass of the particle is constant, and δWtotal is the total work done on the particle, which we obtain by summing the work done by each applied force. From Newton's second law, we can show that

<math>\delta W_{\rm total} = \delta T</math>,

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

<math>T = {m |\mathbf{v}|^2 \over 2}</math>.

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

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 V:

<math>\mathbf{F} = - \nabla V</math>.

Suppose all the forces acting on a particle are conservative, and V is the total potential energy, obtained by summing the potential energies corresponding to each force. Then

<math>\mathbf{F} \cdot \delta \mathbf{r} = - \nabla V \cdot \delta \mathbf{r} = - \delta V</math>
<math>\Rightarrow - \delta V = \delta T</math>
<math>\Rightarrow \delta (T + V) = 0</math>.

This result is known as the conservation of energy, and states that the total energy, <math>E = T + V</math>, is constant in time. It is often useful, because most commonly encountered forces are conservative.

Further results

Newton's laws provide many important results for composite bodies. See angular momentum.

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.

History

To Be Done

See also

Edmund Halley -- List of equations in classical mechanics

Further Reading

  • Feynman, R., Six Easy Pieces.
  • ---, Six Not So Easy Pieces.
  • ---, Lectures on Physics.
  • Kleppner, D. and Kolenkow, R. J., An Introduction to Mechanics, McGraw-Hill (1973).



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