basic_tools:dimensional_analysis
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basic_tools:dimensional_analysis [2017/10/22 17:25] jakobadmin [Examples] |
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| ====== Dimensional Analysis ====== | ====== Dimensional Analysis ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | Dimensional analysis is an extremely powerful tool that allows us to derive the solution for many complicated systems or equations, without doing any actual calculations. | ||
| - | For example, complicated Gaussian integrals can be "calculated" using dimensional analysis, up to some constant, using solely dimensional analysis. Another example, is the speed of an apple at the ground after being dropped from some height, or the frequency of a pendulum. For more details, see the "Examples" tab. | + | <tabbox Intuitive> |
| - | + | ||
| - | In addition, dimensional analysis is one of the most important tools in [[quantum_theory:quantum_field_theory|quantum field theory]]. | + | |
| - | <tabbox Layman> | + | |
| <note tip> | <note tip> | ||
| Line 13: | Line 8: | ||
| </note> | </note> | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| **Recommended Resources:** | **Recommended Resources:** | ||
| - | * A brilliant discussion with many great examples can be found in chapter 1 of Mahajan's book "Stree-fighting Mathematics". | + | * A brilliant discussion with many great examples can be found in chapter 1 of Mahajan's book "[[https://mitpress.mit.edu/sites/default/files/titles/content/9780262514293_Creative_Commons_Edition.pdf|Street-fighting Mathematics]]". |
| * https://particlephd.wordpress.com/2008/12/08/dimensional-analysis-for-animals/ | * https://particlephd.wordpress.com/2008/12/08/dimensional-analysis-for-animals/ | ||
| - | * | + | * [[https://philpapers.org/rec/LANDE|Dimensional Explanations]] by Marc Lange |
| + | |||
| **Dimensional Analysis in Quantum Field Theory:** | **Dimensional Analysis in Quantum Field Theory:** | ||
| - | see http://math.ucr.edu/home/baez/renormalizability.html for how dimensional analysis is used in [[quantum_theory:quantum_field_theory|quantum field theory]] and also: | + | see http://math.ucr.edu/home/baez/renormalizability.html for how dimensional analysis is used in [[theories:quantum_field_theory:canonical|quantum field theory]] and also: |
| <blockquote>Looking at the Lagrangian density in (1), we can easily work out what the units | <blockquote>Looking at the Lagrangian density in (1), we can easily work out what the units | ||
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| <cite>[[http://www.physics.rutgers.edu/~zrwan/thesis/physics/Towards-Final-Laws.txt|Towards the final laws of physics]] by Steven Weinberg</cite></blockquote> | <cite>[[http://www.physics.rutgers.edu/~zrwan/thesis/physics/Towards-Final-Laws.txt|Towards the final laws of physics]] by Steven Weinberg</cite></blockquote> | ||
| + | ---- | ||
| - | + | **Examples** | |
| - | <tabbox Researcher> | + | |
| - | **Recommended Resources:** | ||
| - | * [[http://www.sciencedirect.com/science/article/pii/0003491681900725|Dimensional Analysis in field theory]] by P.M Stevenson | ||
| - | |||
| - | |||
| - | |||
| - | |||
| - | --> Common Question 1# | ||
| - | |||
| - | |||
| - | <-- | ||
| - | |||
| - | --> Common Question 2# | ||
| - | |||
| - | |||
| - | <-- | ||
| - | | ||
| - | <tabbox Examples> | ||
| --> Speed of an apple after being dropped from some height h# | --> Speed of an apple after being dropped from some height h# | ||
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| across a change of fundamental units of measurement.** The dimensions for the quantities involved in our problem are the following: | across a change of fundamental units of measurement.** The dimensions for the quantities involved in our problem are the following: | ||
| - | $$ [\theta] = T; quad [l] = L; \quad [m]= M; \quad [g]=LT^{-2} . $$ | + | $$ [\theta] = T; \quad [l] = L; \quad [m]= M; \quad [g]=LT^{-2} . $$ |
| Now, consider the quantity $l/g$. If the unit of length is decreased by a factor | Now, consider the quantity $l/g$. If the unit of length is decreased by a factor | ||
| Line 144: | Line 124: | ||
| dimensionless. In the jargon of dimensional analysis, we have "nondimensionalized" the problem. | dimensionless. In the jargon of dimensional analysis, we have "nondimensionalized" the problem. | ||
| - | In principle, $\Pi$ depends (just like $\theta$ under our guess) upon the quantities $l, | + | In principle, $\Pi$ depends (just like $\theta$ under our guess) upon the quantities $l$, |
| - | m, and g: $\Pi = \Pi(l,m,g)$. If we decrease the unit of mass by some factor c, of | + | $m$, and $g$: $\Pi = \Pi(l,m,g)$. If we decrease the unit of mass by some factor c, of |
| course, the numerical value for mass will increase by that same factor c. But, | course, the numerical value for mass will increase by that same factor c. But, | ||
| - | in so doing, neither n nor l nor g will change in value. In particular, $\Pi(l,m,g)$ | + | in so doing, neither $\Pi$ nor $l$ nor $g$ will change in value. In particular, $\Pi(l,m,g)$ |
| - | is independent of m. What happens to $\Pi$ if we decrease the unit of length by | + | is independent of $m$. What happens to $\Pi$ if we decrease the unit of length by |
| some factor a leaving the unit of time unchanged? While the value for length | some factor a leaving the unit of time unchanged? While the value for length | ||
| - | will increase by a factor of a, the quantity $\Pi$ , as it is dimensionless, remains | + | will increase by a factor of $a$, the quantity $\Pi$ , as it is dimensionless, remains |
| unchanged. Hence, $\Pi(l,m,g)$ is independent of $l$. Finally, what happens to $\Pi$ if we decrease the unit of time by a factor of $b$ while leaving the unit of length | unchanged. Hence, $\Pi(l,m,g)$ is independent of $l$. Finally, what happens to $\Pi$ if we decrease the unit of time by a factor of $b$ while leaving the unit of length | ||
| invariant. We have seen that this results in the numerical value for acceleration | invariant. We have seen that this results in the numerical value for acceleration | ||
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| <-- | <-- | ||
| + | |||
| + | <tabbox Abstract> | ||
| + | |||
| + | **Recommended Resources:** | ||
| + | |||
| + | * [[http://www.sciencedirect.com/science/article/pii/0003491681900725|Dimensional Analysis in field theory]] by P.M Stevenson | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | Dimensional analysis is an extremely powerful tool that allows us to derive the solution for many complicated systems or equations, without doing any actual calculations. | ||
| + | |||
| + | For example, complicated Gaussian integrals can be "calculated" using dimensional analysis, up to some constant, using solely dimensional analysis. Another example, is the speed of an apple at the ground after being dropped from some height, or the frequency of a pendulum. For more details, see the "Examples" tab. | ||
| + | |||
| + | In addition, dimensional analysis is one of the most important tools in [[theories:quantum_field_theory:canonical|quantum field theory]]. | ||
| | | ||
| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
basic_tools/dimensional_analysis.1508685912.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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