变分是在应用数学与变分法中泛函应对与函数中的微分使用的概念。具体可以分为泛函的变分、函数的变分等。[1] 设极值曲线为 y ^ = y ^ ( x ) {\displaystyle {\hat {y}}={\hat {y}}(x)} ，可取曲线为 y = y ( x ) {\displaystyle y=y(x)} 。定义 δ y = y ^ − y {\displaystyle \delta {y}={\hat {y}}-y} 为y的一次变分，即函数y的增量。从而可得 δ y ′ = y ^ ′ − y ′ {\displaystyle \delta {y'}={\hat {y}}'-y'} 对隐函数 φ ( x , y ) = 0 {\displaystyle \varphi (x,y)=0} ，其一次变分即为全微分： δ φ = δ y ∂ φ ∂ y + δ x ∂ φ ∂ x {\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}+\delta x{\frac {\partial {\varphi }}{\partial {x}}}} 。由于x无增量，即 δ x = 0 {\displaystyle \delta x=0} ，故有 δ φ = δ y ∂ φ ∂ y {\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}} 。 对泛函 min y J ( y ) = ∫ x 0 x 1 F ( x , y ( x ) , y ′ ( x ) ) d x {\displaystyle {\underset {y}{\min }}{J(y)}=\int _{x_{0}}^{x_{1}}F(x,y(x),y'(x))dx} ， 可得 J ( y ^ ) − J ( y ) = ∫ x 0 x 1 ( ∂ F ∂ y δ y + ∂ F ∂ y ′ δ y ′ ) d x + O ( δ y ) {\displaystyle J({\hat {y}})-J(y)=\int _{x_{0}}^{x_{1}}({\frac {\partial F}{\partial y}}\delta y+{\frac {\partial F}{\partial y'}}\delta y')dx+O(\delta y)} ，其一次变分是其Taylor级数的一次项，即 δ J = ∫ x 0 x 1 ( ∂ F ∂ y δ y + ∂ F ∂ y ′ δ y ′ ) d x {\displaystyle \delta J=\int _{x_{0}}^{x_{1}}({\frac {\partial F}{\partial y}}\delta y+{\frac {\partial F}{\partial y'}}\delta y')dx} ，或直接定义一次变分为 δ J ( y , h ) = d d ε J ( y + ε h ) | ε = 0 {\displaystyle \delta J(y,h)={\frac {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{\varepsilon =0}\,} 。 故其二次变分为其Taylor级数的二次项，即 δ 2 J = 1 2 ∫ x 0 x 1 ( ∂ 2 F ∂ y 2 ( δ y ) 2 + ∂ 2 F ∂ y ∂ y ′ δ y δ y ′ + ∂ 2 F ∂ y ′ 2 ( δ y ′ ) 2 ) d x {\displaystyle \delta ^{2}J={\frac {1}{2}}\int _{x_{0}}^{x_{1}}({\frac {\partial ^{2}F}{\partial y^{2}}}(\delta y)^{2}+{\frac {\partial ^{2}F}{\partial y\partial y'}}\delta y\delta y'+{\frac {\partial ^{2}F}{\partial y'^{2}}}(\delta y')^{2})dx} 。 需要注意，与二阶微分 d 2 y = d ( d y ) {\displaystyle d^{2}y=d(dy)} 不同，泛函的二次变分不是对其一次变分再取变分。 实例 计算 J ( y ) = ∫ a b y y ′ d x {\displaystyle J(y)=\int _{a}^{b}yy'dx\,} 的一次变分？ δ J ( y , h ) {\displaystyle \delta J(y,h)\,} = d d ε J ( y + ε h ) | ε = 0 {\displaystyle ={\frac {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{\varepsilon =0}} = d d ε ∫ a b ( y + ε h ) ( y ′ + ε h ′ ) d x | ε = 0 {\displaystyle ={\frac {d}{d\varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ dx\left.\right|_{\varepsilon =0}} = d d ε ∫ a b ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ ) d x | ε = 0 {\displaystyle ={\frac {d}{d\varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\left.\right|_{\varepsilon =0}} = ∫ a b d d ε ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ ) d x | ε = 0 {\displaystyle =\int _{a}^{b}{\frac {d}{d\varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\left.\right|_{\varepsilon =0}} = ∫ a b ( y h ′ + y ′ h + 2 ε h h ′ ) d x | ε = 0 {\displaystyle =\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ dx\left.\right|_{\varepsilon =0}} = ∫ a b ( y h ′ + y ′ h ) d x {\displaystyle =\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ dx} 泛函 变分法 控制论 脚注 [1]吴, 受章. 最优控制理论与应用. 外链 Exampleproblems.com（页面存档备份，存于互联网档案馆）有更多计算泛函一次变分的实例。Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for Firefox
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，其一次变分是其Taylor级数的一次项，即
，或直接定义一次变分为 
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不同，泛函的二次变分不是对其一次变分再取变分。
