Classical mechanics is a model of 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 models objects at rest), kinematics (which models objects in motion), and dynamics (which models objects subjected to forces). See also mechanics.
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 It is superseded by relativistic mechanics for systems moving at large velocities near the speed of light, quantum mechanics for systems at small distance scales, and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 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 discovered in the late 19th century that can only be resolved by more modern physics. In particular, classical nonrelativistic electrodynamics predicts that the speed of light is a constant relative to an aether medium, 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 black body is predicted to emit infinite amounts of energy. The effort at resolving these problems led to the development of quantum mechanics.
Description of the theory
The following introduces the basic concepts of classical mechanics. For simplicity, it uses 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 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 properly described by quantum mechanics. Objects with non-zero size have more complicated behavior than hypothetical point particles, because their internal configuration can change - 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. Such composite objects behave like point particles, provided they are small compared to the distance scales of the problem, which indicates that the use of point particles is internally 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. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames.
The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or
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
- 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
The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time or
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 speed of u to S.
- v'' = v - u (the velocity of a particle from the perspective of S is slower by u than 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
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
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, 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:
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
This can be integrated to obtain
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.
If a force F is applied to a particle that achieves a displacement δr, the work done by the force is the scalar quantity
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:
where T is called the kinetic energy. For a point particle, it is defined as
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 V:
If 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
This result is known as conservation of energy and states that the total energy, E = T + V, is constant in time. It is often useful, because many commonly encountered forces are conservative.
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. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.
Consider two reference frames, one of which is traveling at a relative speed of u to the other. For example, for a car passing another car at a relative speed of 10 km/h, u is 10 km/h.
Two reference frames S and S' , with S' traveling at a relative speed of u to S; an event has space-time coordinates of (x,y,z,t) in S and (x' ,y' ,z' ,t' ) in S'.
The space-time coordinates of an event in Galilean-Newtonian relativity are governed by the set of formulas which defines a group transformation known as the Galilean transformation:
Assuming time is considered an absolute in all reference frames, the relationship between space-time coordinates in reference frames differing by a relative speed of u in the x direction (let x = ut when x' = 0) is:
- x' = x - ut
- y' = y
- z' = z
- t' = t
The set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform).
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 also developed the calculus which is necessary to perform the mathematical calculations involved in classical mechanics.
After Newton the field became more mathematical and more abstract.
- Feynman, Richard Phillips, Six Easy Pieces. ISBN 0201408252
- Feynman, Richard Phillips, and Roger Penrose, Six Not So Easy Pieces. March 1998. ISBN 0201328410
- Feynman, Richard Phillips, Lectures on Physics. ISBN 0738200921
- Kleppner, D. and Kolenkow, R. J., An Introduction to Mechanics, McGraw-Hill (1973). ISBN 0070350485
- Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics (SICM), MIT Press (2001). ISBN 0-262-019455-4
- Rosu, Haret C., "Classical Mechanics (http://arxiv.org/abs/physics/9909035)". Physics Education. 1999. [arxiv.org : physics/9909035]
- Horbatsch, Marko, "Classical Mechanics Course Notes (http://www.yorku.ca/marko/PHYS2010/index.htm)".
- Sussman, Gerald Jay & Wisdom, Jack (2001). Structure and Interpretation of Classical Mechanics (http://mitpress.mit.edu/SICM/)
- Hoiland, Paul (2004). Preferred Frames of Reference & Relativity (http://doc.cern.ch//archive/electronic/other/ext/ext-2004-126.pdf)