Trajetória ortogonal

Uma equação da forma $f(x,y)=c$ , onde $c$ é uma constante, define uma família de curvas. As trajetórias ortogonais são outra família de curvas que intersetam a primeira família em forma ortogonal: em cada ponto de uma das curvas da primeira família passa uma curva da segunda família, formando um ângulo de 90 $^{\circ }$ .^[1]

Para encontrar a família de trajetórias ortogonais às curvas $f(x,y)=c$ , é necessário encontrar uma equação diferencial cuja solução geral seja $f(x,y)=c$ ; essa equação encontra-se derivando implicitamente a equação anterior.

${\partial f \over \partial x}+{\dfrac {\partial f}{\partial y}}{dy \over dx}=0\qquad \Longrightarrow \qquad {dy \over dx}=-$ ${\dfrac {\partial f}{\partial x}} \over {\dfrac {\partial f}{\partial y}}$

A derivada $dy/dx$ representa em cada ponto o declive da curva que passa por esse ponto. O declive da curva ortogonal será o inverso, com sinal trocado

${dy \over dx}=$ ${\dfrac {\partial f}{\partial y}} \over {\dfrac {\partial f}{\partial x}}$

a solução geral desta equação é a família de trajetórias ortogonais.^[1]

Exemplo

Encontre as trajetórias ortogonais da família de círculos com centro na origem.^[1]

Família de círculos com centro na origem e trajetórias ortogonais.

A equação dos círculos com centro na origem é

$x^{2}+y^{2}=c^{2}$

onde o parâmetro $c$ pode ter qualquer valor positivo a equação diferencial cuja solução geral é essa família de círculos obtém-se por derivação implícita

$2x+2yy'=0\qquad \Longrightarrow \qquad {dy \over dx}=-{x \over y}$

e a equação diferencial das trajetórias ortogonais é

${dy \over dx}={\frac {y}{x}}$

A solução desta equação de variáveis separáveis é

$y=ax$

que corresponde a uma família de retas que passam pela origem; a constante de integração é declive das retas. A figura mostra a família de curvas e as trajetórias ortogonais.^[1]

Referências

↑ ^a ^b ^c ^d Villate, Jaime E. (2011). Equações Diferenciais e Equações de Diferenças (PDF). Porto: [s.n.] 120 páginas. Consultado em 13 de julho de 2013

Ver também

v d e Equações diferenciais
Sistema de equações diferenciais	Equações diferenciais · Equações diferenciais ordinárias · Equação diferencial ordinária de primeira ordem · Equação diferencial de d'Alembert · Equação diferencial de Bernoulli · Equação de Clairaut · Equação de Duffing · Decaimento exponencial · Equação de Euler · Equação de Lane-Emden · Equação de Lotka-Volterra · Equação do pêndulo · Equação de Riccati · Crescimento Demográfico · Trajetória ortogonal
Equações diferenciais parciais	Equação de Mason-Weaver · Equação do transporte · Equação de Laplace · Equação de Poisson · Equação do calor · Equações de Navier-Stokes · Equação da onda
Métodos analíticos	Separação de variáveis · Método de Frobenius · Método de d'Alembert
Métodos numéricos	Método dos elementos de contorno · Método dos elementos finitos · Método de Euler · Método Multigrid · Método de Runge-Kutta · Método de Dormand-Prince · Método Wronskiano · Método da variação de parâmetros
Pessoas	Isaac Newton · Leonhard Euler · Charles Émile Picard · Josef Hoëné-Wronski · Ernst Leonard Lindelöf · Rudolf Lipschitz · Augustin-Louis Cauchy · John Crank · Phyllis Nicolson · Carl Runge · Martin Wilhelm Kutta
Outros	Lema de Grönwall · Teorema de Picard-Lindelöf · Teoria de Sturm-Liouville

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