basic_tools:variational_calculus
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| Variational calculus is the alternative to the usual calculus methods when we want to find functions that minimize something. As an analogy, usually when we search for the extrema of a function, we differentiate the function, set the derivative of the function to zero and find the point that yields the extrema. Similar results can be computed by using variational calculus.In variational calculus we find extrema of functionals which are functions of functions with respect some function (instead of variable). This is extremely important for the [[frameworks:lagrangian_formalism|Lagrangian formalism]]. | Variational calculus is the alternative to the usual calculus methods when we want to find functions that minimize something. As an analogy, usually when we search for the extrema of a function, we differentiate the function, set the derivative of the function to zero and find the point that yields the extrema. Similar results can be computed by using variational calculus.In variational calculus we find extrema of functionals which are functions of functions with respect some function (instead of variable). This is extremely important for the [[frameworks:lagrangian_formalism|Lagrangian formalism]]. | ||
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| + | In the same way that the to find the extrema of a function one needs to solve a system of algebraic equation, the result of a [[basic_tools:variational_calculus:functional_derivative|variational derivative]] | ||
| + | is a system of differential equations, these being ordinary or partial differential equations, depending on the function space being search. | ||
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| + | <tabbox A note on notation> | ||
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| + | <blockquote> | ||
| + | The best tool of a physicist is notation abuse. | ||
| + | <cite>Angry Mathematician</cite> | ||
| + | </blockquote> | ||
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| + | One has to be careful. When talking about variational calculus in this section, we need to be in the same page on some things. This is the most important thing to do when starting studying different topics. | ||
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| + | Symbols that appear in this section, by (quasi-)order of apparence | ||
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| + | $$\Omega, q, \mathcal Q, (\cdot), [\cdot], L, q(x), q'(x), T^{(n)} \Gamma, | ||
| + | $$ | ||
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| + | We should learn the meaning of these symbols to keep going. | ||
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| + | There is the main problem, the multiplicity of use of $q$. This can mean: | ||
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| + | * For 1-D dynamics or particle systems it is can be the path $q:\mathbb R\to \mathcal Q$, a point or coordinates of that point of $\mathcal Q$. THey can be vectors or scalars, time dependent or not... | ||
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| + | * For n-D (or field mechanics), q will be usually substitute by a greek leter, and they are not paths, but surfaces or volumes, functions of more than one variable. | ||
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| + | For this reason we will focus on 1-D variational problems for now, but with minimal change on notation we can have the same results for multiple dimensional systems. | ||
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basic_tools/variational_calculus.txt · Last modified: 2021/04/17 19:03 by cleonis
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