essary Condition 例17.1 (due to Galileo Galilei) 在重力作用下,一质点从(x0,3).点沿平面曲线y(x) 无摩擦地自由下滑到(x1-,y) 点,则所需要的时间 (x1y1) d aoo)√2g(30-y 1+ dx o√29(y0-y) 就是y(x)的泛函 这里,要求y(x)一定通过端点(x0,30)和(x1,)
Extremum of a Functional Isoperimetric Problem Conceptual Idea of Functional Extremum of a Functional Extremum of Functional: Necessary Condition Variation Operation ~17.1 (due to Galileo Galilei) 3å^e§:l(x0, y0):÷²¡y(x) à Þ/gdew(x1, y1) :§K¤Im T = Z (x1,y1) (x0,y0) ds p 2g(y0 − y) = Z x1 x0 p 1 + y 02 p 2g(y0 − y) dx Ò´y(x)¼ ùp§¦y(x)½ÏLà:(x0, y0)Ú(x1, y1) C. S. Wu 1Ôù C©{ÐÚ
essary Condition 例17.2弦的横振动问题 设在弦上隔离出足够短的一段弦,则该段弦的 动能==p△ 势能=-T△x 其中α(x,)是弦的横向位移,p是弦的线密度: T是张力 这样,弦的 Hamilton作用量 也是位移(x,的泛函
Extremum of a Functional Isoperimetric Problem Conceptual Idea of Functional Extremum of a Functional Extremum of Functional: Necessary Condition Variation Operation ~17.2 uîįK 3uþ lÑv áãu§KTãu ÄU= 1 2 ρ∆x ∂u ∂t 2 ³U= 1 2 T∆x ∂u ∂x2 Ù¥u(x, t)´uî £§ρ´uݧ T´Üå ù§uHamilton^þ S = Z t1 t0 dt Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx ´ £u(x, t)¼ C. S. Wu 1Ôù C©{ÐÚ
essary Condition 例17.2弦的横振动问题 设在弦上隔离出足够短的一段弦,则该段弦的 动能==p△ 势能=-T△x 其中α(x,)是弦的横向位移,卩是弦的线密度, T是张力 这样,弦的 Hamilton作用量 1 P T d 2"(ot 也是位移v(x,t)的泛函
Extremum of a Functional Isoperimetric Problem Conceptual Idea of Functional Extremum of a Functional Extremum of Functional: Necessary Condition Variation Operation ~17.2 uîįK 3uþ lÑv áãu§KTãu ÄU= 1 2 ρ∆x ∂u ∂t 2 ³U= 1 2 T∆x ∂u ∂x2 Ù¥u(x, t)´uî £§ρ´uݧ T´Üå ù§uHamilton^þ S = Z t1 t0 dt Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx ´ £u(x, t)¼ C. S. Wu 1Ôù C©{ÐÚ
essary Condition 例172弦的横振动问题 au T d at L=/()一(m) 称为 Lagrange量( Lagrangian),而被积函数 称为 Lagrange量密度
Extremum of a Functional Isoperimetric Problem Conceptual Idea of Functional Extremum of a Functional Extremum of Functional: Necessary Condition Variation Operation ~17.2 uîįK S = Z t1 t0 dt Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx L = Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx ¡Lagrangeþ(Lagrangian) § ȼê 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i ¡LagrangeþÝ C. S. Wu 1Ôù C©{ÐÚ
essary Condition 例172弦的横振动问题 au T d at L=/()一(m) 称为 Lagrange量( Lagrangian),而被积函数 du T at 称为 Lagrange量密度
Extremum of a Functional Isoperimetric Problem Conceptual Idea of Functional Extremum of a Functional Extremum of Functional: Necessary Condition Variation Operation ~17.2 uîįK S = Z t1 t0 dt Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx L = Z x1 x0 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i dx ¡Lagrangeþ(Lagrangian) § ȼê 1 2 h ρ ∂u ∂t 2 − T ∂u ∂x2 i ¡LagrangeþÝ C. S. Wu 1Ôù C©{ÐÚ