็ฐกๅŒ–่ฎŠๅˆ†่กจ็คบๆณ•

ๅ‰ไธ€้ƒจๅˆ†ไป‹็ดนไบ†็ฐกๅ–ฎ็š„่ฎŠๅˆ†ๆณ•ๆฆ‚ๅฟต๏ผŒไฝ†ๆ˜ฏ้œ€่ฆๅผ•ๅ…ฅไปปๆ„็š„ๅ‡ฝๆ•ธ\(\eta (t)\)๏ผŒๆ‰‹ๆณ•ไธŠ็จๅซŒ็…ฉ็‘ฃ๏ผŒไธๅˆฉๆ–ผๅพŒ็บŒๆ“ไฝœใ€‚็ฌฌไบŒ้ƒจๅˆ†ไปฅ็›ธๅŒ็š„ๆฆ‚ๅฟต๏ผŒๆŽก็”จๆฏ”่ผƒๆŠฝ่ฑก็š„ๆƒณๆณ•ไฝ†็›ธๅŒ็š„ๆ•ธๅญธๆ‰‹ๆณ•๏ผŒๆผ”็คบไธ€ๆฌก่ฎŠๅˆ†ๆณ•็š„ๆ“ไฝœใ€‚ๆœ‰้ปžๅƒๅฐ‡่ฎŠๅˆ†็š„ๆ“ไฝœ้กžๅŒๆ–ผๅพฎๅˆ†ๆ“ไฝœใ€‚

ๅพฎๅˆ†่ˆ‡่ชคๅทฎ้ …็š„ๅ‡ฝๆ•ธๅœ–ๅƒๅ‡ฝๆ•ธๅœ–ๅƒ็คบๆ„ๅœ–๏ผŒ้กฏ็คบๆ“พๅ‹•ๅพŒ็š„ๅ‡ฝๆ•ธ่ฎŠๅŒ–

้‡ๅฐๅŒไธ€็จฎๆณ›ๅ‡ฝ๏ผˆFunctional๏ผ‰็š„็ฉๅˆ†ๅ•้กŒ $$S[F(x,\dot{x},t)]=\int_a^b F(x,\dot{x},t)dt $$ ็•ถๆˆ‘ๅ€‘้‡ๅฐ\(x\)ๅš่ฎŠๅˆ†๏ผŒ $$xโ†’x+\delta x$$ ่ฎŠๅˆ†\(\delta x\)ๆปฟ่ถณ $$\delta x(a)=\delta x(b)=0$$ \(x\)็š„่ฎŠๅˆ†ๆœƒๅฐŽ่‡ด\(\dot{x}\)ใ€\(F\)็™ผ็”Ÿ่ฎŠๅŒ– $$ \dot{x} (x) \to \dot{x} (x+\delta x) =\dot{x}(x)+\delta \dot{x}$$ $$F(x,\dot{x},t)โ†’F(x+\delta x,\dot{x}(x)+\delta \dot{x},t)=F(x,\dot{x},t)+\delta F $$ ๆˆ‘ๅ€‘่ฆ\(\delta S=0\) $$\delta S=\delta \int _a^b Fdt=\int _a^b \delta Fdt=0$$ ๆฆ‚ๅฟตไธŠๅพˆๅฅฝ็†่งฃ็š„ๆ˜ฏ๏ผŒ\(\delta F\)็š„่ฎŠๅŒ–ๅ’Œ\(\delta x\)ใ€\(\delta \dot{x}\)ๆœ‰้—œ๏ผŒๆ‰€ไปฅๅฐ‡\(\delta F\)ๅฑ•้–‹ $$\delta S=\int _a^b \delta Fdt=\int _a^b { \partial F\over\partial x }\delta x+{\partial F\over \partial \dot{x} } \delta \dot{x} dt $$

้€ฒ้šŽ๏ผšThm.1 :ๅพฎๅˆ†่ˆ‡่ฎŠๅˆ†ๅฐ่ชฟ
ๅฆ‚ๆžœไปŠๅคฉๅพฎๅˆ†่ˆ‡่ฎŠๅˆ†้‡ๅฐ็š„ๅฐ่ฑกไธๅŒ๏ผŒๅฆ‚ๅฐ\(t\)ๅพฎๅˆ† \(d \over dt\)ใ€ๅฐ\(x\)่ฎŠๅˆ†\(\delta x\)๏ผŒๅ‰‡ \(d \over dt\)่ˆ‡\(\delta \)ๅฏไปฅๅฐ่ชฟใ€‚ $$\delta \dot{f}=\dot{f} (x+\delta x)-\dot{f}(x)$$ $$= {d \over dt} \left(f(x+\delta x)-f(x)\right)= {d \over dt} \delta f$$

ๅฐ‡\(\delta \dot{x}\)ๅฐ่ชฟ \({d \over dt} \delta x\)
$$\delta S=\int _a^b {\partial F\over\partial x} \delta xdt+ \int _a^b {\partial F\over \partial \dot{x} } \left( {d \over dt} \delta x\right)dt $$ ๅŒๆจฃ็š„ๆ‰‹ๆณ•ๅฐ็ฌฌไบŒ้ …ๅšๅˆ†้ƒจ็ฉๅˆ† $$\int _a^b {\partial F\over \partial \dot{x} } \left( {d \over dt} \delta x\right)dt = \color{red}{{\partial F\over \partial \dot{x} } \delta x \Big|_a^b}-\int _a^b {d \over dt} {\partial F\over \partial \dot{x} } \delta xdt$$ ๆณจๆ„็ด…่‰ฒ้€™ไธ€้ …๏ผŒๅ› ็‚บๆˆ‘ๅ€‘่ฆๆฑ‚่ฎŠๅˆ†\(\delta x\)ๆปฟ่ถณ\(\delta x(a)=\delta x(b)=0\)๏ผŒๆ‰€ไปฅ\( \color{red}{{\partial F\over \partial \dot{x} } \delta x\Big|_a^b=0}\) $$\delta S=\int _a^b {\partial F\over\partial x} \delta xdt-\int _a^b \left( {d \over dt} {\partial F\over \partial \dot{x} } \right)\delta xdt $$ $$=\int _a^b \left( {\partial F\over\partial x}- {d \over dt} {\partial F\over \partial \dot{x} } \right)\delta xdt=0$$ \(\delta x \) is arbitrary. $$ {\partial F\over\partial x}- {d \over dt} {\partial F\over \partial \dot{x} } =0$$ Euler-Lagrange equation

้€ฒ้šŽ๏ผšThm.2:่ฎŠๅˆ†็š„Chain rule $$\delta (FG)={\partial (FG)\over\partial x} \delta x+{\partial (FG)\over \partial \dot{x} } \delta \dot{x}$$ $$=\left( {\partial F\over\partial x} G+F {\partial G\over\partial x}\right)\delta x+\left( {\partial F\over \partial \dot{x} } G+F {\partial G\over\partial \dot{x}} \right)\delta \dot{x}$$ $$=\left( {\partial F\over\partial x} \delta x+ {\partial F\over \partial \dot{x} } \delta \dot{x} \right)G+F\left({\partial G\over\partial x} \delta x+{\partial G\over\partial \dot{x}} \delta \dot{x} \right)$$ $$= \delta F\cdot G+F\cdot \delta G$$


้€ฒ้šŽ๏ผšThm.3:้‡ๅฐๅ‡ฝๆ•ธFๅŒไน˜ๅŒ้™คๅฆไธ€ๅ‡ฝๆ•ธG๏ผŒไธๅฝฑ้Ÿฟ่ฎŠๅˆ† $$\delta F=\delta \left(F\cdot {G\over G}\right)=\delta \left(F\cdot G\cdot G^{-1} \right)$$ $$=\delta F\cdot G\cdot G^{-1}+F\cdot \delta G\cdot G^{-1}+F\cdot G\cdot \delta (G^{-1} )$$ $$=\delta F+F\cdot \delta G\cdot G^{-1}+F\cdot G\cdot \left(-{\delta G\over G^2} \right)$$ $$=\delta F$$

็Ž‹ๅŸนๅ„’ ๆœ€ๅฐไฝœ็”จ้‡ๅŽŸ็†๏ผˆๅคๅ…ธ๏ผ‰