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Euler’s Equation Find the Extremum Define a function along a trajectory. • • • • y(a,x) = y(0,x) + ah(x) Parametric function Variation h(x) is C1 function. End points h(x1) = h(x2) = 0 x2 y(a, x) y(x) Find the integral J • If y is varied J must increase x1 J f ( y( x), y' ( x); x)dx x1 x2 Parametrized Integral Write the integral in parametrized form. Condition for extremum J f ( y(a , x), y' (a , x); x)dx x1 x2 J a 0 a 0 for all h(x) Expand with the chain rule • Term a only appears with h x2 f y J f y ( )dx x1 y a a y a x2 f J f dh ( h ( x) )dx x1 y a y dx Apply integration by parts … Euler’s Equation x2 x1 x2 ( x2 d f dh f f x )dx h ( x ) x2 ( )h ( x)dx 1 x1 dx y y dx y h(x1) = h(x1) = 0 x1 x2 d f dh f ( )dx ( )h ( x)dx x1 dx y y dx x2 f x2 f J d f d f h ( x) y h ( x)dx x1 y dx y h ( x)dx x1 a dx y f d f 0 y dx y It must vanish for all h(x) This is Euler’s equation Geodesic A straight line is the shortest distance between two points in Euclidean space. Curves of minimum length are geodesics. • Tangents remain tangent as they move on the geodesic • Example: great circles on the sphere Euler’s equation can find the minimum path. Soap Film Find a surface of revolution. • Find the area • Minimize the function y (x2, y2) f x 1 y 2 f d f d xy 0 y 0 dx y dx 1 y 2 xy a 2 1 y y a x a 2 2 (x1, y1) dA 2 dx 2 dy 2 x2 x1 y b x a cosh a A 2 x 1 y2 dx Action Motion involves a trajectory in configuration space Q. • Tangent space TQ for full description. The integral of the Lagrangian is the action. S L(q j , q j ; t )dt t1 t2 Find the extremum of action • Euler’s equation can be applied to the action • Euler-Lagrange equations Q q’ q L d L j 0 j q dt q next

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Euler's equation, Euler's formula, Euler equation, Complex numbers, Euler's identity, the fluid, complex number, exponential function, Taylor series, imaginary parts

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posted: | 11/25/2009 |

language: | English |

pages: | 7 |

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