Mathematical models in physics
Encyclopedia
Mathematical models are of great importance in physics
. Physical theories
are almost invariably expressed using mathematical model
s, and the mathematics involved is generally more complicated than in the other sciences. Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry
is much used in classical physics, while special relativity
and general relativity
are examples of theories that use geometries
which are not Euclidean.
It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box
are among the many simplified models used in physics.
Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits relativity theory and quantum mechanics
must be used, even these do not apply to all situations and need further refinement. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduce to Newtonian mechanics when the speed much less than the speed of light
. Quantum mechanics reduce to classical physics when the quantum numbers are high. If we say that a tennis ball is a particle and calculate its de Broglie wavelength it will turn out to be insignificantly small so it is seen that classical physics is better to use than quantum mechanics in this case.
The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations
and the Schrödinger equation
. These laws are such as a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital
models that are approximate solutions to the Schrödinger equation. In engineering
, physics models are often made by mathematical methods such as finite element analysis.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. Physical theories
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
are almost invariably expressed using mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s, and the mathematics involved is generally more complicated than in the other sciences. Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
is much used in classical physics, while special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
are examples of theories that use geometries
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
which are not Euclidean.
It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
are among the many simplified models used in physics.
Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits relativity theory and quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
must be used, even these do not apply to all situations and need further refinement. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduce to Newtonian mechanics when the speed much less than the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Quantum mechanics reduce to classical physics when the quantum numbers are high. If we say that a tennis ball is a particle and calculate its de Broglie wavelength it will turn out to be insignificantly small so it is seen that classical physics is better to use than quantum mechanics in this case.
The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
and the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. These laws are such as a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
models that are approximate solutions to the Schrödinger equation. In engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, physics models are often made by mathematical methods such as finite element analysis.
See also
- Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
- Differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s - ProbabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...