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 It is superseded by relativistic mechanics for systems moving at large velocities near the speed of light, quantum mechanics for systems at small distance scales, and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 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 blackbody 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

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

.

The acceleration, or rate of change of velocity, is

.

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

.

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, 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

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

.

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 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

.

Suppose the mass of the particle is constant, and δW_total 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

δW_total = δT,

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 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:

.

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

.

This result is known as the conservation of energy, and states that the total energy, E = T + V, 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.

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.

History

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 also performed the famous experiment of dropping two canon balls from the tower of Pisa (The theory, and the practice showed that they both hit the ground at the same time).

Sir Isaac Newton was the first to propose the three laws of motion (the law of inertia, the 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.

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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Thermodynamics is the study of energy, its conversions between various forms such as heat, and the ability of energy to do work. It is closely related to statistical mechanics from which many themodynamic relationships can be derived.

It can be argued that thermodynamics was misnamed as it does not actually relate to rates of change as such and therefore would probably have been better called thermostatics as a field. Thermodynamics relates to whether certain chemical reactions are possible but not how quickly they occur.

The field covers a wide range of topics including, but not limited to: efficiency of heat engines and turbines, phase equilibria, PVT relationships. gas laws (both ideal and non ideal), energy balances, heats of reactions, and combustion reactions. It is governed by 4 basic laws (in brief):

The Laws of Thermodynamics

Alternative statements are given for each law. These statements are, for the most part, mathematically equivalent.

• Zeroth law: A fundamental concept within thermodynamics, however, it was not termed a law until after the first three laws were already widely in use, hence the zero numbering. There is some discussion about its status. Stated as:

  • If each of two systems is in thermal equilibrium with a third system, all must be in equilibrium with each other.

• 1st Law: Is stated as follows:

  • Energy can neither be created nor destroyed only changed.

  • The heat flowing into a system equals the sum of change in internal energy plus the work done by the system.

    • The work exchanged in an adiabatic process depends only on the initial and the final state and not on the details of the process.

    • The sum of heat flowing into a system and work done by the system is zero.

• 2nd Law: A far reaching and powerful law, it can be stated many ways, the most popular of which is:

  • It is impossible to obtain a process such that the unique effect is the subtraction of a positive heat from a reservoir and the production of a positive work.

    • A system operating in contact with a thermal reservoir cannot produce positive work in its surroundings (Kelvin)

    • A system operating in a cycle cannot produce a positive heat flow from a colder body to a hotter body (Clausius)

  • The entropy of a closed system never decreases (see Maxwell's demon)

• 3rd Law: This law explains why it is so hard to cool something to absolute zero:

  • All processes cease as temperature approaches zero.

  • As temperature goes to 0, the entropy of a system approaches a constant

The three original laws have been humorously summarised as: (1) you can't win; (2) you can't break even; (3) you can't get out of the game.

Basics

This is a brief summary and collection of the major concepts in thermodynamics. To learn more about each, just click on the corresponding links:

U stands for the internal energy, T stands for temperature, S stands for entropy, P stands for pressure, V stands for volume, ρ stands for density, F stands for Helmholtz free energy, H stands for enthalpy, G stands for Gibbs free energy, μ stands for chemical potential and N stands for particle number.

The rest of this discussion is about systems in equilibrium only. For nonequilibrium thermodynamics, see ...

Substances describable by temperature alone

Blackbody radiation is an example. The reason why this is the case is because photon number isn't conserved. The state is completely described by its temperature except at phase transitions and perhaps spontaneous symmetry breaking in the ordered phase. given the internal energy as a function of temperature, we can define F=U-TS.

Substances describable by temperature and pressure alone

Most "pure" nonmagnetic substances fall into this category. This state is completely described by its temperature and pressure, except at phase transitions and perhaps spontaneous symmetry breaking in the ordered phase. Given U and V (or the density ρ) as a function of T and P, we can define the Helmholtz energy as before and the Gibbs energy as G=U-TS+PV and the enthalpy as H=U+PV.

Substances describable by temperature, pressure and chemical potential

If there are more than one kind of atom/molecule, a substance would fall into this category. This state is completely described by its temperature, pressure and chemical potentials, except at phase transitions and perhaps spontaneous symmetry breaking in the ordered phase.

Substances describable by temperature and magnetic field

If a substance is a ferromagnet or a superconductor, for example, it would fall into this category. It is completely described by its temperature and magnetic field, except at phase transitions and perhaps spontaneous symmetry breaking in the ordered phase.

Thermodynamic Systems

A thermodynamic system is that part of the universe that is under consideration. A real or imaginary boundary separates the system from the rest of the universe, which is referred to as the surroundings. Often thermodynamic systems are characterized by the nature of this boundary as follows:

• Isolated systems are completely isolated from their surroundings. Neither heat nor matter can be exchanged between the system and the surroundings. An example of an isolated system would be an insulated container, such as an insulated gas cylinder. (In reality, a system can never be absolutely isolated from its environment, because there is always at least some slight coupling, even if only via minimal gravitational attraction).

• Closed systems are separated from the surroundings by an impermeable barrier. Heat can be exchanged between the system and the surroundings, but matter cannot. A greenhouse is an example of a closed system.

• Open systems can exchange both heat and matter with their surroundings. Portions of the boundary between the open system and its surroundings may be impermeable and/or adiabatic, however at least part of this boundary is subject to heat and mass exchange with the surroundings. The ocean would be an example of an open system.

Thermodynamic State

A key concept in thermodynamics is the state of a system. When a system is at equilibrium under a given set of conditions, it is said to be in a definite state. For a given thermodynamic state, many of the system's properties have a specific value corresponding to that state. The values of these properties are a function of the state of the system and are independent of the path by which the system arrived at that state. The number of properties that must be specified to describe the state of a given system is given by Gibbs phase rule. Since the state can be described by specifying a small number of properties, while the values of many properties are determined by the state of the system, it is possible to develop relationships between the various state properties. One of the main goals of Thermodynamics is to understand these relationships between the various state properties of a system. Equations of State are examples of some of these relationships.

Thermodynamics also touches upon the fields of:

• Fluid mechanics

• Calorimetry

• Thermal Analysis

• Thermochemistry also known as chemical thermodynamics

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Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of Mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in every day life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum). In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.

At the heart of statistical mechanics is the partition function:

where k is Boltzmann's constant, T is the temperature and E_i reflects each possible energetic state of the system. This is the version for systems which don't allow an exchange of matter.

The partition function provides a measure of the total number of energetic states available to the system at a given temperature.

It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes.

A partition function can be defined for each mode. Simple expressions have been derived relating each of the various modes to various measurable molecular properties, such as the characteristic rotational or vibrational frequencies.

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Electromagnetism is a theory unified by James Clerk Maxwell to explain the interrelationship between electricity and magnetism. At the heart of this theory is the notion of an electromagnetic field.

A stationary electromagnetic field stays bound to its origin. Examples of stationary fields are: the magnetic field around a wire carrying current or the electric field between the plates of a capacitor.

A changing electromagnetic field propagates away from its origin in the form of a wave. These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves, microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle physics this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.

The subfield of electromagnetism dealing specifically with the rapidly changing electric and magnetic fields which constitute light, is called electrodynamics.

The whole of electromagnetism is governed by Maxwell's equations, which are compatible with and served as a motivation for the theory of relativity.

Electromagnetic Method

A geophysical method in which the magnetic and or electric fields resulting from generated surface currents are measured. Measurements may be made in the frequency domain at a number of frequencies, or the time domain at several time intervals after a transient pulse. Natural field methods such as magnetotellurics (MT) use natural magnetic and electromagnetic field as the source.

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Light cone

The special theory of relativity (SR) is the physical theory published in 1905 by Albert Einstein that modified Newtonian physics to incorporate electromagnetism as represented by Maxwell's equations. The theory is called "special" because the theory applies only to the special case of measurements made when both the observer and that which is being observed are not affected by gravity. Ten years later, Einstein published the theory of General Relativity, or GR for short, which is the extension of special relativity to incorporate gravitation.

Motivation for the theory of special relativity

Before the formulation of special relativity, Hendrik Lorentz and others had already noted that electromagnetics differed from Newtonian physics in that observations by one of some phenomenon can differ from those of a person moving relative to that person at speeds nearing the speed of light. For example, one may observe no magnetic field, yet another observes a magnetic field in the same physical area. Lorentz suggested an aether theory in which objects and observers travelling with respect to a stationary aether underwent a physical shortening (Lorentz-Fitzgerald contraction) and a change in temporal rate (time dilation). This allowed the partial reconciliation of electromagnetics and Newtonian physics. When the velocities involved are much less than speed of light, the resulting laws simplify to Newton's laws. The theory, known as Lorentz Ether Theory (LET) was criticized (even by Lorentz himself) because of its ad hoc nature.

While Lorentz suggested the Lorentz transformation equations as a mathematical description that accurately described the results of measurements, Einstein's contribution was to derive these equations from a more fundamental theory. Einstein wanted to know what was invariant (the same) for all observers. His original title for his theory was (translated from German) "Theory of Invariants". It was Max Planck who suggested the term "relativity" to highlight the notion of transforming the laws of physics between observers moving relative to one another.

Special relativity is usually concerned with the behaviour of objects and observers which remain at rest or are moving at a constant velocity. In this case, the observer is said to be in an inertial frame of reference or simply inertial. Comparison of the position and time of events as recorded by different inertial observers can be done by using the Lorentz transformation equations. A common misstatement about relativity is that SR cannot be used to handle the case of objects and observers who are undergoing acceleration (non-inertial reference frames), but this is incorrect. For an example, see the relativistic rocket problem. SR can correctly predict the behaviour of accelerating bodies as long as the acceleration is not due to gravity, in which case general relativity must be used.

Invariance of the speed of light

SR postulated that the speed of light in vacuum is the same to all inertial observers, and said that every physical theory should be shaped or reshaped so that it is the same mathematically for every inertial observer. This postulate (which comes from Maxwell's equations for electromagnetics) together with the requirement, successfully reproduces the Lorentz transformation equations, and has several consequences that struck many people as bizarre, among which are:

• The time lapse between two events is not invariant from observer to another, but is dependent on the relative speeds of the observers' reference frames.

• The twin paradox is the "story" of a twin who flies off in a spaceship travelling near the speed of light. When he returns he discovers that his twin has aged much more rapidly than he has (or he aged more slowly).

• Two events that occur simultaneously in different places in one reference frame may occur one after the other in another reference frame (relativity of simultaneity).

• The dimensions (e.g. length) of an object as measured by an observer may differ from those by another.

• The mass of a particle increases as it's velocity increases. This led to the famous equation E = mc². See below.

Lack of an absolute reference frame

Another radical consequence is the rejection of the notion of an absolute, unique, frame of reference. Previously it had been suggested that the universe was filled with a substance known as "aether" (absolute space), against which speeds could be measured. Aether had some wonderful properties: it was sufficiently elastic that it could support electromagnetc waves, those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, culminating in the famous Michelson-Morley experiment, suggested that either the Earth was always stationary, or the notion of an absolute frame of reference was mistaken and must be discarded.

Equivalence of mass and energy

Perhaps most far reaching, it also showed that energy and mass, previously considered separate, were equivalent, and related by the most famous expression from the theory:

E = mc²

where E is the energy of the body (at rest), m is the mass and c is the speed of light.

The most practical implication of this theory is that it puts an upper limit to the laws (see Law of nature) of Classical Mechanics and gravity formed by Isaac Newton at the speed of light. Nothing carrying mass can move faster than this speed. As an object's velocity approaches the speed of light, the amount of energy required to accelerate it approaches infinity, making it impossible to reach the speed of light. Only particles with no mass, such as photons, can actually achieve this speed (and in fact they must always travel at this speed in all frames of reference), which is approximately 300,000 kilometers per second or 186,300 miles per second.

The name "tachyon" has been used for hypothetical particles which would move faster than the speed of light, but to date evidence of the actual existence of tachyons has not been produced.

Simultaneity

Special relativity also holds that the concept of simultaneity is relative to the observer: A 'time-like interval' has end-points separated by a path along which it is possible for a hypothetical matter or light to travel. A 'space-like interval' has end-points separated by a path in space-time along which neither light nor any slower-than-light signal could travel. No information can pass between points separated by a space-like interval. Events along a space-like interval cannot influence one another by transmitting light or matter, and would appear simultaneous to an observer in the right frame of reference. To observers in different frames of reference, event A could seem to come before event B or vice-versa; this does not apply to events separated by time-like intervals.

Status of Special Relativity

Special relativity is now universally accepted by the physics community, unlike General Relativity which is still insufficiently confirmed by experiment to exclude certain alternative theories of gravitation. However, there are a handful of people opposed to relativity on various grounds and who have proposed various alternatives, mainly Aether theories. One alternative theory is doubly-special relativity, where a characteristic length is added to the list of invariant quantities.

The Geometry of Space-time in Special Relativity

SR uses a 'flat' 4 dimensional space, usually referred to as space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.

Tests of postulates of special relativity

• Michelson-Morley experiment - ether drift

• Hamar experiment - obstruction of ether flow

• Trouton-Noble experiment - torque on a capacitor

• Kennedy-Thorndike experiment - time contraction

• Forms of the emission theory experiment

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Video: Visualization of Einstein's special relativity