拉普拉斯—龍格—楞次矢量

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本文鏈接：拉普拉斯—龍格—楞次矢量（排版更漂亮，敘述更清晰）

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預備知識 開普勒問題

在開普勒問題中，我們定義拉普拉斯—龍格—楞次矢量（Laplace-Runge-Lenz Vector） （通常簡稱為 LRL 矢量）為

$ewcommand{I}{mathrm{i}} ewcommand{E}{mathrm{e}} enewcommand{vec}[1]{oldsymbol{mathbf{#1}}} ewcommand{mat}[1]{oldsymbol{mathbf{#1}}} ewcommand{ en}[1]{oldsymbol{mathbf{#1}}} ewcommand{Nabla}{vec abla} enewcommand{Tr}{^{mathrm{T}}} ewcommand{uvec}[1]{hat{oldsymbol{mathbf{#1}}}} enewcommand{pmat}[1]{egin{pmatrix}#1end{pmatrix}} ewcommand{ali}[1]{egin{aligned}#1end{aligned}} ewcommand{leftgroup}[1]{left{egin{aligned}#1end{aligned} ight.} ewcommand{vmat}[1]{egin{vmatrix}#1end{vmatrix}} ewcommand{Cj}{^*} ewcommand{Her}{^dagger} ewcommand{Q}[1]{hat #1} ewcommand{sinc}{mathop{mathrm{sinc}}} ewcommand{Si}[1]{,mathrm{#1}} ewcommand{les}{leqslant} ewcommand{ges}{geqslant} ewcommand{qty}[1]{left{{#1} ight}} ewcommand{qtyRound}[1]{left({#1} ight)} ewcommand{qtySquare}[1]{left[{#1} ight]} ewcommand{abs}[1]{leftlvert{#1} ight vert} ewcommand{eval}[1]{left.{#1} ight vert} ewcommand{comm}[2]{left[{#1},{#2} ight]} ewcommand{commStar}[2]{[{#1},{#2}]} ewcommand{pb}[2]{left{{#1},{#2} ight}} ewcommand{pbStar}[2]{{{#1},{#2}}} ewcommand{vdot}{oldsymbolcdot} ewcommand{cross}{oldsymbol imes} ewcommand{Nabla}{vec abla} ewcommand{grad}{oldsymbol abla} ewcommand{div}{vec ablaoldsymbolcdot} ewcommand{curl}{vec ablaoldsymbol imes} ewcommand{laplacian}{ abla^2} ewcommand{sinRound}[2][{}]{sin^{#1}left(#2 ight)} ewcommand{cosRound}[2][{}]{cos^{#1}left(#2 ight)} ewcommand{ anRound}[2][{}]{ an^{#1}left(#2 ight)} ewcommand{cscRound}[2][{}]{csc^{#1}left(#2 ight)} ewcommand{secRound}[2][{}]{sec^{#1}left(#2 ight)} ewcommand{cotRound}[2][{}]{cot^{#1}left(#2 ight)} ewcommand{sinhRound}[2][{}]{sinh^{#1}left(#2 ight)} ewcommand{coshRound}[2][{}]{cosh^{#1}left(#2 ight)} ewcommand{ anhRound}[2][{}]{ anh^{#1}left(#2 ight)} ewcommand{arcsinRound}[2][{}]{arcsin^{#1}left(#2 ight)} ewcommand{arccosRound}[2][{}]{arccos^{#1}left(#2 ight)} ewcommand{arctanRound}[2][{}]{arctan^{#1}left(#2 ight)} ewcommand{expRound}[1]{expleft(#1 ight)} ewcommand{logRound}[2][{}]{log^{#1}left(#2 ight)} ewcommand{lnRound}[2][{}]{ln^{#1}left(#2 ight)} enewcommand{Re}{mathrm{Re}} enewcommand{Im}{mathrm{Im}} ewcommand{dd}[1][]{,mathrm{d}^{#1}} ewcommand{dv}[2][{}]{frac{mathrm{d}^{#1}}{mathrm{d}{#2}^{#1}}} ewcommand{dvStar}[2][{}]{mathrm{d}^{#1}/mathrm{d}{#2}^{#1}} ewcommand{dvTwo}[3][{}]{frac{mathrm{d}^{#1}{#2}}{mathrm{d}{#3}^{#1}}} ewcommand{dvStarTwo}[3][{}]{mathrm{d}^{#1}{#2}/mathrm{d}{#3}^{#1}} ewcommand{pdv}[2][{}]{frac{partial^{#1}}{partial{#2}^{#1}}} ewcommand{pdvStar}[2][{}]{partial^{#1}/partial{#2}^{#1}} ewcommand{pdvTwo}[3][{}]{frac{partial^{#1}{#2}}{partial{#3}^{#1}}} ewcommand{pdvStarTwo}[3][{}]{partial^{#1}{#2}/partial{#3}^{#1}} ewcommand{pdvThree}[3]{frac{partial^2{#1}}{partial{#2}partial{#3}}} ewcommand{pdvStarThree}[3]{partial^2{#1}/partial{#2}partial{#3}} ewcommand{ra}[1]{leftlangle{#1} ight vert} ewcommand{raStar}[1]{langle{#1} vert} ewcommand{ket}[1]{leftlvert{#1} ight angle} ewcommand{ketStar}[1]{lvert{#1} angle} ewcommand{raket}[1]{leftlangle{#1}middle|{#1} ight angle} ewcommand{raketStar}[1]{langle{#1}|{#1} angle} ewcommand{raketTwo}[2]{leftlangle{#1}middle|{#2} ight angle} ewcommand{raketStarTwo}[2]{langle{#1}|{#2} angle} ewcommand{ev}[1]{leftlangle{#1} ight angle} ewcommand{evStar}[1]{langle{#1} angle} ewcommand{evTwo}[2]{leftlangle{#2}middle|{#1}middle|{#2} ight angle} ewcommand{evStarTwo}[2]{langle{#2}|{#1}|{#2} angle} ewcommand{mel}[3]{leftlangle{#1}middle|{#2}middle|{#3} ight angle} ewcommand{melStar}[3]{langle{#1}|{#2}|{#3} angle} ewcommand{order}[1]{mathcal{O}left(#1 ight)} egin{equation} vec A = vec p cross vec L - mk uvec r end{equation}$

$ewcommand{I}{mathrm{i}} ewcommand{E}{mathrm{e}} enewcommand{vec}[1]{oldsymbol{mathbf}} ewcommand{mat}[1]{oldsymbol{mathbf{#1}}} ewcommand{ en}[1]{oldsymbol{mathbf{#1}}} ewcommand{Nabla}{vec abla} enewcommand{Tr}{^{mathrm{T}}} ewcommand{uvec}[1]{hat{oldsymbol{mathbf{#1}}}} enewcommand{pmat}[1]{egin{pmatrix}#1end{pmatrix}} ewcommand{ali}[1]{egin{aligned}#1end{aligned}} ewcommand{leftgroup}[1]{left{egin{aligned}#1end{aligned} ight.} ewcommand{vmat}[1]{egin{vmatrix}#1end{vmatrix}} ewcommand{Cj}{^*} ewcommand{Her}{^dagger} ewcommand{Q}[1]{hat #1} ewcommand{sinc}{mathop{mathrm{sinc}}} ewcommand{Si}[1]{,mathrm{#1}} ewcommand{les}{leqslant} ewcommand{ges}{geqslant} ewcommand{qty}[1]{left{{#1} ight}} ewcommand{qtyRound}[1]{left({#1} ight)} ewcommand{qtySquare}[1]{left[{#1} ight]} ewcommand{abs}[1]{leftlvert{#1} ight vert} ewcommand{eval}[1]{left.{#1} ight vert} ewcommand{comm}[2]{left[{#1},{#2} ight]} ewcommand{commStar}[2]{[{#1},{#2}]} ewcommand{pb}[2]{left{{#1},{#2} ight}} ewcommand{pbStar}[2]{{{#1},{#2}}} ewcommand{vdot}{oldsymbolcdot} ewcommand{cross}{oldsymbol imes} ewcommand{Nabla}{vec abla} ewcommand{grad}{oldsymbol abla} ewcommand{div}{vec ablaoldsymbolcdot} ewcommand{curl}{vec ablaoldsymbol imes} ewcommand{laplacian}{ abla^2} ewcommand{sinRound}[2][{}]{sin^{#1}left(#2 ight)} ewcommand{cosRound}[2][{}]{cos^{#1}left(#2 ight)} ewcommand{ anRound}[2][{}]{ an^{#1}left(#2 ight)} ewcommand{cscRound}[2][{}]{csc^{#1}left(#2 ight)} ewcommand{secRound}[2][{}]{sec^{#1}left(#2 ight)} ewcommand{cotRound}[2][{}]{cot^{#1}left(#2 ight)} ewcommand{sinhRound}[2][{}]{sinh^{#1}left(#2 ight)} ewcommand{coshRound}[2][{}]{cosh^{#1}left(#2 ight)} egin{equation} mathbf A = mathbf p cross mathbf L - mk uvec r end{equation}$

其中為質點動量，為軌道角動量，，為質點位矢的單位矢量．在開普勒問題中，可以證明是一個守恆量．

守恆證明

　　我們下面證明．對式 1 求時間導數，考慮到中心力場中質點角動量 LL 守恆，有

其中由牛頓第二定律和萬有引力定律，有

˙ $dot{vec p} = vec F = - frac{k} {r^2} vec r$

又由「極坐標中單位矢量的偏導」得

最後由式 5，極坐標系中的角動量等於

$egin{equation} vec L = mr^2dot heta vec z end{equation}$

將式 3至式 5代入式 2得

$egin{equation} dot{vec A} = -frac{k}{r^2}vec r imes (mr^2dot hetavec z) - mkdot hetavec heta =-mkdot heta (vec r imes vec z + vec heta) = vec 0 end{equation}$