“Euler’s equation”的概念、定义、翻译、参考文献-科学参考

Euler’s equation

Mathematics

A central result in the calculus of variations which addresses problems such as finding curves of minimal length (geodesics) or solving the brachistochrone problem. A functional I associates a real number

$I (y) = \int_{a}^{b} F (x, y, y^{'}) d x$

(where a, b are fixed limits) with each of a family of graphs y(x); this functional might represent the arc length of such graphs, the time taken to travel along the graphs under gravity, etc. Euler’s equation then states I is maximal when

$\frac{d}{d x} (\frac{\partial F}{\partial y'}) - \frac{\partial F}{\partial y} = 0.$

As an example, the arc length of a curve y(x) from (a,c) to (b,d) is

$I (y) = \int_{a}^{b} \sqrt{1 + y^{'} {(x)}^{2}} d x,$

so that $F (x, y, y^{'}) = \sqrt{1 + {y′}^{2}}$ and Euler’s equation then reads

$\frac{d}{d x} (\frac{y'}{\sqrt{1 + {y′}^{2}}}) - 0 = 0 .$

Integrating and rearranging we find that y′ is constant, and so the graph of y is a straight line.

单词	Euler’s equation
释义	Euler’s equation Mathematics A central result in the calculus of variations which addresses problems such as finding curves of minimal length (geodesics) or solving the brachistochrone problem. A functional I associates a real number $I (y) = \int_{a}^{b} F (x, y, y^{'}) d x$ (where a, b are fixed limits) with each of a family of graphs y(x); this functional might represent the arc length of such graphs, the time taken to travel along the graphs under gravity, etc. Euler’s equation then states I is maximal when $\frac{d}{d x} (\frac{\partial F}{\partial y'}) - \frac{\partial F}{\partial y} = 0.$ As an example, the arc length of a curve y(x) from (a,c) to (b,d) is $I (y) = \int_{a}^{b} \sqrt{1 + y^{'} {(x)}^{2}} d x,$ so that $F (x, y, y^{'}) = \sqrt{1 + {y′}^{2}}$ and Euler’s equation then reads $\frac{d}{d x} (\frac{y'}{\sqrt{1 + {y′}^{2}}}) - 0 = 0 .$ Integrating and rearranging we find that y′ is constant, and so the graph of y is a straight line.
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