Dimensional analysis

n mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, (denoted M, L, T, Q, and Θ, respectively). For example, velocity, which may be measured in meters per second (m/s), miles per hour (mi/h), or some other units, has dimension L/T.

Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their "dimensions", or their lack thereof.

The basic principle of dimensional analysis was known to Isaac Newton (1686) who referred to it as the "Great Principle of Similitude"^[1]. Important contributions were made by the 19th century French mathematician Joseph Fourier,^[2] based on the idea that the physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.

Contents

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• 1 Introduction
  • 1.1 Definition
  • 1.2 Mathematical properties
  • 1.3 Mechanics
  • 1.4 Other fields of physics and chemistry
  • 1.5 Commensurability
  • 1.6 Polynomials and transcendental functions
  • 1.7 Incorporating units
  • 1.8 Position vs displacement
  • 1.9 Orientation/Frame of reference
  • 1.10 Other uses
• 2 Examples
  • 2.1 A simple example: period of a harmonic oscillator
  • 2.2 A more complex example: energy of a vibrating wire
• 3 Extensions
  • 3.1 Huntley's extension: directed dimensions
  • 3.2 Siano's extension: orientational analysis
• 4 Percentages and derivatives
• 5 Dimensionless
  • 5.1 Dimensionless constants
  • 5.2 Dimensionless theories
• 6 Applications
  • 6.1 Mathematics
  • 6.2 Finance, economics, and accounting
• 7 See also
• 8 Notes
• 9 References
• 10 External links