User:Maschen/Equation (physics)

Equation (physics)

Physics includes a very number of formulae, there are too many to all of them list in any one place. Equations can be written using different formalisms, constants and variables can be substituted to obtain alternative formulae (for those which serve as a substitute). There are enough to classify them into groups in a number of ways typically with crossing boundaries, as they should since physics is a unity.

Introduction
The numerous formulae which can be classified in numerous ways. One of them is given in the scheme below. Of course boundarys can overlap, since an equation might serve as a law or definition, and be a starting point or an end product of a mathematical argument.

Since physics is a highly quantitative natural science, it is naturally outlined and constructed by mathematics. Physical quantities are often tensors; most usually scalars and vectors. In advanced branches more complex mathematical objects like spinors and twistors are used.

Physical quantities can be related by equations, equations can be ordinary equalities between expressions of quantities, be functions of any necessary number of variables, differential/integral equations, in advanced fields include operators and functionals.

Their purpose is to determine one physical quantity from other known ones, or make predictions of the behaviour of a system. They can also serve as lemmata to further develop the physical theory or result in useful end products, i.e. corollaries.

Sometimes the mathematical armature of a physical theory initiates with defining equations, any known fundamental laws, conservation laws, and continuity equations (which follow from ), and through mathematical mixing and manipulation of definitions with laws, new equations are generated. Equations can be written in multiple forms using defining equations.

Laws
Laws and principles

A law or principle (usually interchangeable terms) is a statement which summarizes related ideas and observations for a collection of physical phenomena, as accurately as possible and to the extent of no known exceptions.

Laws may be considered absolute if they always seem to describe naturally occurring physical phenomena exactly as described by the law, or in the last resort they may be approximate and applicable to a limiting situation, requiring a more accurate law. Due to the perception problem of physics (see Philosophy of perception), there is no way to tell how exact laws are in nature.

They can generally be summarized by a single equation, or sometimes a small collection of equations. Typically they are very simple.

Universal laws

A universal law is one which has no exception in the universe, it can always be applied no matter what event in space or time. They would generally be fundamental, since their range of validity is independent on space-time for the situation in question.

Fundamental laws

A fundamental law is one which has no foundation, because it serves as a foundation for other derived laws. They cannot be derived, but are used to derive other results.

Empirical laws

Laws derived from experimental observation and facts.

Derived laws

Laws which are not fundamental, they can be derived.

Theorem

A formal and proven statement which is not a law, but still used for deriving other results. They may or may not be fundamental.

Rule

Rules are simply systematic procedures/algorithms for simplifying a calcluation. They may or may not have a theoretical foundation, i.e. any rigourous reason as to why they work, they simply happen to. They are not fundamental at all.

Conservation and continuity
Conservation laws

An effect/occurrence which has a property/quantity possessing a form of symmetry, homogeneity, and invariance is conserved between any two events in space-time. The effect is summarized by a conservation law. They are often fundamental, but not always. Some may be approximate, or applicable only in a limiting situation.

Continuity equations

Following from conservation laws are continuity equations which describes the transport of a conserved quantity. In a local region the quantity may not be conserved, so loss/gain of the quantity may occur in that region due to variations of production from local sources or sinks.

Definitions, constructed relations
Defining equation

An equation which defines a physical quantity in terms of others (not necessarily others).

Derived Equations

Equation derived from other theorems, laws, definitions etc.

Constitutive equations

An equation between physical quantities characteristic to matter, or sometimes to a vacuum, which approximates the response of that material/substance/vacuum to external influences.

Results
Lemma and lemmata

A lemma is a proven statement used to continue to a larger result. They are not used as stand-alone theorems. A lemmata is a lemma which leads the way forward in a derivation in a number of directions.

Corollaries

A result which follows directly from a more general one, i.e. a special case. Sometimes more than one corollaries can follow from a general result.

Specialization/Usefulness

Sometimes more simpler classifications may be used/added on:


 * Useful Results
 * Important Results
 * Specilized Results