Classical mechanics is the most common system of physics in use today. It is the physics of 'ordinary' situations, considering objects too large to exhibit quantum effects, too slow to exhibit relativistic effects, and not dense enough to require general relativity. Specifically, it is a deterministic formulation of mechanics with absolute time. It retains its utility in a relativistic, quantum universe because, in its domain of application, it is a powerful and accurate theory.

Newtonian Dynamics

The simpler of the two formulations of classical mechanics is Newtonian dynamics. The basis of Newtonian dynamics is Newton's Three Laws of Motion. The most important of these for dynamics is the Second Law; defining force, F, as mass times acceleration. This connects the dynamical world of forces and energies with the kinematical world of accelerations and positions. The expression works both ways; given a force you can find the acceleration, and given the acceleration you can find out what the force must be.

Several derived quantities are used in Newtonian dynamics. The simplest is momentum, the product of mass and velocity. Work is the dot product of a force and a displacement, and is the change in energy over that displacement. Power is the rate of performance of work or rate of change of energy, and is often calculated as the dot product of the force on a body and its velocity. The most important derived quantity is energy.

Energy comes in two separate forms in classical mechanics, potential energy and kinetic energy. Kinetic energy is the simplest of the two, it is just a scalar representation of the motion of the body, and is equal to half of the product of mass and velocity squared. Potential energy is a more complicated concept, best explained in terms of conservative forces. A conservative force is a force which does the same amount of work on a body moving between two points for all possible paths between those two points. It is proven in advanced calculus that for a force field with that property a scalar function can be defined such that the force is its gradient. This scalar function is the potential energy. The sum of the potential energy and the kinetic energy is the total energy.

Conservation laws are, as in all physics, vitally important in Newtonian dynamics. The main conservation laws used in Newtonian dynamics are conservation of energy, conservation of momentum, and conservation of angular momentum. These laws are used to simplify the mathematics of closed dynamical systems and allow them to be solved more simply than by directly considering the forces and accelerations of the bodies. All of these conservation laws can also be adapted for cases where there are external forces, i.e. the system is not closed. The external force is the rate of change of the momentum, and the external torque is the rate of change of the angular momentum. The work done on the system between two states is the change in the energy between those states. Despite this, it is preferable, if the external force is conservative, to include it in the system as another source of potential energy.

Lagrangian Dynamics

An alternate formulation of classical dynamics was derived by Joseph Lagrange. This formulation simplifies the mathematics by exploiting the difference between conservative force and forces of constraint. Forces of constraint are forces that link together elements of the system and constrain their movement. In the Newtonian formulation, forces of constraint can often be awkward to deal with. If the system has n coordinates and m forces of constraint, it has n - m degrees of freedom. The Lagrangian formulation incorporates the forces of constraint by defining (n - m) generalized coordinates, denoted by the symbol q.

An example of generalised coordinates is in a system consisting of two masses connected by a (massless) string over a pulley. The string provides a force of constraint between the two masses. Under the Newtonian formulation, the tension in the string must be included in the formulae. The Lagrangian formulation defines a generalised coordinate as the distance of one of the masses from the pulley. The position of the other mass is uniquely determined by this coordinate because the string has a constant length. The generalised coordinates can be differentiated to find the generalised velocities, q'i, and the generalised coordinates and generalised velocities together comprise the independent variables of the Lagrangian formulation

The key to the Lagrangian formulation is a quantity called the Lagrangian, L. The Lagrangian is defined as the kinetic energy minus the potential energy. The Lagrangian is important because the integral of the Lagrangian over a path, the action, is minimised along the path that will actually be taken. This principle, the action principle, can be converted into a more useful form, Lagrange's Equations, which state that the time derivative of the derivative of the Lagrangian with respect to a generalised velocity is equal to the derivative of the Lagrangian with respect to the corresponding generalised coordinate. An exhaustive overview of the Lagrangian formulation can be found in Purvis's writeup at Lagrangian Mechanics.

Hamiltonian Dynamics

Yet another formulation of classical mechanics was developed by William R. Hamilton. The Hamiltonian formulation is closely related to the Lagrangian formulation, but approaches the problem from a different perspective. It is not as commonly used as the Lagrangian formulation in classical mechanics, but Schrodinger's equation of quantum mechanics is related to the Hamiltonian formulation.

One of the major differences between the Lagrangian and Hamiltonian formulations is their choice of independent variable. Lagrange's equations use generalised velocities and generalised coordinates. Hamilton's equations replace the generalised velocities with the generalised momenta, pi, which are defined as the derivative of the Lagrangian with respect to the generalised velocity. A complete statement of Hamiltonian dynamics can be found at Hamiltonian.

Relation to Other Theories

Classical mechanics was the precursor to both quantum mechanics and special relativity, and as such is an approximation to both theories. Special relativity was created to reconcile mechanics and electrodynamics, which in their classical incarnations were inconsistent; classical electrodynamics predicted that the speed of light is constant, and classical mechanics required that it vary depending on the relative motion of the light and reference frame. In other words, classical mechanics predicts that you can 'catch up' to a beam of light, which under relativity is found to be impossible. The equations of classical mechanics can be retrieved from the equations of special relativity by taking the limit as the ratio v/c approaches zero.

Quantum mechanics is more significantly divorced from the basic equations of classical mechanics than special relativity is. The strange behaviour of quantum systems seems to defy any sort of classification in classical terms. Nevertheless, the correspondence principle of quantum mechanics predicts that as the quantum numbers get large, the behaviour of the system should approach the classical (or relativistic) prediction. This is required because everything in the 'classical' domain of macroscopic objects and slow speeds is made up of quantum constituents. If the equations of quantum mechanics did not converge to the classical results as the scale increases, they would be inconsistent with the real world, which obeys the classical result to considerable accuracy in 'ordinary' circumstances.

Conclusion

Classical mechanics is the oldest comprehensive physical theory which accurately reflects the nature of reality in a particular domain. Beginning with the work of Newton, and expanded usefully by the work of Lagrange, it describes the mechanical universe in the domain where speeds are much smaller than the speed of light and the energy scale is much larger than Planck's constant. Since most of the perceptible universe is at these scales, classical mechanics is applied to many problems, including rockets, stars, engines, and air pucks. For these situations and many more, classical physics is all you need.


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This writeup is copyright 2003-2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .

Wine and whiskey, gin and meade
Always helped me to succeed
At physics, like a stumbling ass whom providence has smiled upon.

And so I came to doubt my worth and wonder at the other folk
Whose careful thought and patience earned them less than my own pencil play.

I did as other wisemen had
When similarly irked:
Another glass of bourbon,
And I got on with my work.

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