1626435102-9ccf6d0a03aed727d9bf8add4160ccd9 (Учебное пособие (Головин, Чесноков)), страница 6

Çàôèêñèðóåìα > 0 è ðàññìîòðèì îäíîñòîðîííèå ïðåäåëû â (2.51) ïðè t → t∗ ∓ 0 (â ýòîì ñëó÷àå ïåðåìåííàÿ τ (t) = tan f2t → ±∞). Ôóíêöèÿ f2 arctan(ατ (t)), ðàçðûâíàÿ â òî÷êå t = t∗ , èìååò îäíîñòîðîííèå ïðåäåëû ðàâíûåπ/f ñëåâà è −π/f ñïðàâà îò ðàçðûâà. Ôóíêöèÿ χ(t) ÿâëÿåòñÿ êóñî÷íî-ïîñòîÿííîé è ïðèíèìàåò çíà÷åíèÿ 2πn/f ñëåâàè 2π(n + 1)/f ñïðàâà îò òî÷êè t = t∗ . Ïîýòîìó ïðè t → t∗ ∓ 0 ôóíêöèÿ t(t, α) = f2 arctan(ατ ) + χ(t) èìååò îäíîñòîðîííèå ïðåäåëû, êîòîðûå ñîâïàäàþò è ðàâíû t∗ = (2n + 1)π/f , ÷òî ñîãëàñóåòñÿ ñ ïåðâîé ôîðìóëîé (2.52). Âû÷èñëÿÿîäíîñòîðîííèå ïðåäåëû äëÿ îñòàëüíûõ ôóíêöèé â ôîðìóëàõ (2.51), óáåæäàåìñÿ â èõ ñîâïàäåíèè ñ ñîîòâåòñòâóþùèìèâûðàæåíèÿìè â (2.52).

Òàêèì îáðàçîì, äîîïðåäåëåííûå â òî÷êàõ t = t∗ ôóíêöèè (2.51) ÿâëÿþòñÿ íåïðåðûâíûìè ïîâñåì àðãóìåíòàì.Çàâèñèìîñòü ôóíêöèé r, θ, U , V , h îò ïåðåìåííûõ r, θ, U , V è h ëèíåéíàÿ, ïîýòîìó äèôôåðåíöèðóåìîñòü ïî ýòèìàðãóìåíòàì íå âûçûâàåò ñîìíåíèÿ. Òàêæå î÷åâèäíî, ÷òî ðàññìàòðèâàåìûå ôóíêöèè íåïðåðûâíî äèôôåðåíöèðóåìûïî ïåðåìåííîé t âî âñåõ òî÷êàõ, êðîìå t = t∗ . Äèôôåðåíöèðóÿ ðàññìàòðèâàåìîå îòîáðàæåíèå ïî t è ïåðåõîäÿ ê ïðåäåëóïðè t → t∗ ∓ 0, óáåæäàåìñÿ â ñóùåñòâîâàíèè è ñîâïàäåíèè îäíîñòîðîííèé ïðåäåëîâ:∂t∂t |t=t∗∂U∂t |t=t∗= α1 ,=∂r∂θ∂t |t=t∗ = 0,∂t |t=t∗(α2 −1)f 2 r√, ∂V∂t |t=t∗ = 0,4α α=(α−1)f2α ,∂h∂t |t=t∗= 0.Ïðîâåäåííûé àíàëèç ïîêàçûâàåò, ÷òî îòîáðàæåíèå (2.51), äîîïðåäåëåííîå â òî÷êàõ t∗ = (2n + 1)π/f â ñîîòâåòñòâèè ñôîðìóëàìè (2.52), ÿâëÿåòñÿ íåïðåðûâíî äèôôåðåíöèðóåìûì ïî âñåì àðãóìåíòàì.Òàê êàê ñèñòåìà óðàâíåíèé (2.48) äîïóñêàåò èíôèíèòåçèìàëüíûé îïåðàòîð X̂9 , òî ñîãëàñíî îáùåé òåîðèè óðàâíåíèÿ(2.48) íå ìåíÿþòñÿ ïðè çàìåíå ïåðåìåííûõ, ñîîòâåòñòâóþùåé êîíå÷íûì ïðåîáðàçîâàíèÿì îïåðàòîðà X̂9 .

Ýòî ïîçâîëÿåò ñôîðìóëèðîâàòü ñëåäóþùåå óòâåðæäåíèå, óáåäèòüñÿ â ñïðàâåäëèâîñòè êîòîðîãî ìîæíî è íåïîñðåäñòâåííûìèâû÷èñëåíèÿìè.Óòâåðæäåíèå. Åñëè ñîâîêóïíîñòü ôóíêöèé U = U (t, r, θ), V = V (t, r, θ), h = h(t, r, θ) óäîâëåòâîðÿåò ñèñòåìå34óðàâíåíèé (2.48), òî òîé æå ñèñòåìå óðàâíåíèé óäîâëåòâîðÿåò ñîâîêóïíîñòü ôóíêöèéqf r (α2 −1)τα2 +τ 2U (t, r, θ) = U (t, r, θ) + 2 α2 +τ 2,α(1+τ 2 )q2−α)α2 +τ 2,V (t, r, θ) = V (t, r, θ) − f2r (α−1)(τ22α +τα(1+τ 2 )h(t, r, θ) =ãäå(2.53)α2 +τ 2h(t, r, θ),α(1+τ 2 )τ = tan( f2t ), τ = tan( f2t ), t = f2 arctan(ατ ) + χ(t),q2)r = r α(1+τθ = θ + arctan(τ ) − arctan(ατ ),21+α τ 2 ,α ïðîèçâîëüíîå ïîëîæèòåëüíîå ÷èñëî, χ(t) êóñî÷íî-ïîñòîÿííàÿ ôóíêöèÿ, ðàâíàÿ 2πk/f , íà èíòåðâàëå t ∈√((2k − 1)π/f, (2k + 1)π/f ), k öåëîå.  òî÷êàõ t = (2k + 1)π/f ïîëàãàåì, ÷òî t = t, r = r/ α, θ = θ.Íàéäåííîå ïðåîáðàçîâàíèå ïîçâîëÿåò ñîïîñòàâèòü âñåì èçâåñòíûì òî÷íûì ðåøåíèÿì ðàññìàòðèâàåìîé ìîäåëè íîâûå òî÷íûå ðåøåíèÿ.

Óðàâíåíèÿ (2.48) äîïóñêàþò êëàññ ñòàöèîíàðíûõ âðàùàòåëüíî-ñèììåòðè÷íûõ ðåøåíèéRr V̂ 21dr + h0 ,U = Û = 0, V = V̂ (r), h = ĥ(r) = g(2.54)r + f V̂0ãäå V̂ (r) ïðîèçâîëüíàÿ ãëàäêàÿ ôóíêöèÿ, h0 íåîòðèöàòåëüíàÿ ïîñòîÿííàÿ. Ñîãëàñíî óòâåðæäåíèþ, íàáîð ôóíêöèéU=f r (α2 −1)τ2 1+α2 τ 2 ,q qα(1+τ 2 )V̂ r 1+α2 τ 2 − τ = tan f2tα(1+τ 2 )1+α2 τ 2V = q22)α(1+τ )h = α(1+τ;1+α2 τ 2 ĥ r1+α2 τ 2f r (α−1)(ατ 2 −1),21+α2 τ 2(2.55)òàêæå ÿâëÿåòñÿ ðåøåíèåì ñèñòåìû óðàâíåíèé (2.48), êîòîðîå â òî÷êàõ t = (2n + 1)π/f ñëåäóåò äîîïðåäåëèòü ïîíåïðåðûâíîñòè.Ïóñòü ¾áàçîâîå¿ ðåøåíèå ñîîòâåòñòâóåò ñîñòîÿíèþ ïîêîÿ:Û = 0,V̂ = 0,35ĥ = h0 .Ïîäñòàâëÿÿ ýòè ôóíêöèè â (2.55), ïîëó÷àåì íîâîå òî÷íîå ðåøåíèå, îáëàäàþùåå ñâîéñòâîì ïåðèîäè÷íîñòè ïî âðåìåíè:f r (α2 −1)τ2 1+α2 τ 2 ,U=V =f r (α−1)(ατ 2 −1),21+α2 τ 2h=α(1+τ 2 )1+α2 τ 2 .(2.56)Òðàåêòîðèè äâèæåíèÿ ÷àñòèö îïðåäåëÿþòñÿ èç ñèñòåìû óðàâíåíèédrdtè èìåþò âèär(t) = r0dθdt= U,q=1+α2 τ 21+τ 2 ,Vr,r|t=0 = r0 ,θ|t=0 = θ0θ(t) = θ0 + arctan(ατ ) − arctan(τ ).Ýòè óðàâíåíèÿ ìîæíî ïåðåïèñàòü ñëåäóþùèì îáðàçîì(x − a)2 + (y − b)2 = R2 ,ãäåa=(α+1)r02cos θ0 ,b=(α+1)r02sin θ0 ,R=(α−1)r0.2×àñòèöû æèäêîñòè â ðåøåíèè (2.56) äâèæóòñÿ ïî îêðóæíîñòÿì è çà âðåìÿ T = 2π/f âîçâðàùàþòñÿ â èñõîäíîå ïîëîæåíèå.

Ðåøåíèå îïèñûâàåò ïåðèîäè÷åñêèé ïðîöåññ ðàñòåêàíèÿ/ñæàòèÿ æèäêîãî öèëèíäðà ñ ïîäâèæíîé íåïðîíèöàåìîéãðàíèöåé äâèæóùåéñÿ ïî çàêîíópr = R0(1 + τ 2 )−1 (1 + α2 τ 2 ),ãäå R0 íà÷àëüíûé ðàäèóñ öèëèíäðà.2.6Çàäà÷è1. Âû÷èñëèòü ïåðâîå è âòîðîå ïðîäîëæåíèå îïåðàòîðîâ:(a) X = x2 ∂x + xu∂u ,(b) X = 4t∂t − u∂u ,Z = R2 (t, x) × R(u);Z = R2 (t, x) × R(u);(c) X = ∂t + x∂x + u∂u + 2v∂v ,Z = R2 (t, x) × R2 (u, v);36(d) X = −t∂t + ∂x + u∂u + w∂w + 2v∂v ,Z = R2 (t, x) × R3 (u, w, v);(e) X = (x − y)∂x + (x + y)∂y ,Z = R(x) × R(y);(f) X = (x − y)∂x + (x + y)∂y ,Z = R2 (x, y) × R(z).2.

Ïîêàçàòü, ÷òî óðàâíåíèå íåëèíåéíîé òåïëîïðîâîäíîñòèut − u4 uxx = 0äîïóñêàåò àëãåáðó Ëè L5 îïåðàòîðîâX1 = ∂ t ,X2 = ∂x ,X4 = 4t∂t − u∂u ,X3 = 2t∂t + x∂x ,X5 = x2 ∂x + xu∂u .3. Ïîêàçàòü, ÷òî óðàâíåíèÿ îäíîìåðíîé ãàçîâîé äèíàìèêè äëÿ ïîëèòðîïíîãî ãàçàut + uux + ρ−1 px = 0,ρt + ρux + uρx = 0,pt + γpux + upx = 0äîïóñêàþò àëãåáðó Ëè L6 îïåðàòîðîâX1 = ∂t , X2 = ∂x , X3 = t∂t + x∂x , X4 = t∂x + ∂u ,X5 = t∂t − u∂u + 2ρ∂ρ , X6 = ρ∂ρ + p∂p . ñëó÷àå γ = 3 äîïóñêàåìàÿ àëãåáðà Ëè ðàñøèðÿåòñÿX7 = t2 ∂t + tx∂x + (x − tu)∂u − tρ∂ρ − 3tp∂p .4.

Âû÷èñëèòü ãðóïïû ïðåîáðàçîâàíèé, äîïóñêàåìûõ ÎÄÓ âòîðîãî ïîðÿäêà.(a) y 00 = y −2 y 0 − (xy)−1 ;37(b) y 00 = exp(−y 0 );(c) y 00 = x−4 y(y 0 )3 ;(d) y 00 = y 2 exp(−x);(e) y 00 = xy + y −1 (y 0 )2 .5. Íàéòè ãðóïïó ïðåîáðàçîâàíèé, äîïóñêàåìûõ îäíîìåðíûìè óðàâíåíèÿìè ìåëêîé âîäû (g = const)ut + uux + ghx = 0,ht + hux + uhx = 0.6. Íàéòè ïðåîáðàçîâàíèÿ ðàñòÿæåíèÿ, äîïóñêàåìûå óðàâíåíèåì ïîãðàíè÷íîãî ñëîÿ íà ïîëóáåñêîíå÷íîé ïëàñòèíå(çàäà÷à Áëàçèóñà)ψy ψxy − ψx ψyy = νψyyy ,ψ(x, 0) = ψy (x, 0) = 0,ψy → U (y → ∞).Çäåñü ν è U ïîñòîÿííûå.7.

Äîêàçàòü, ÷òî ìîäåëü äâóìåðíûõ èçýíòðîïè÷åñêèõ äâèæåíèé ïîëèòðîïíîãî ãàçà ñ ïîêàçàòåëåì àäèàáàòû γ = 2,îïèñûâàåìàÿ ñèñòåìîé äèôôåðåíöèàëüíûõ óðàâíåíèéut + uux + vuy + 2ccx = 0,vt + uvx + vvy + 2ccy = 0,ct + ucx + vcy + 2−1 c(ux + vy ) = 0äîïóñêàåò ïðîåêòèâíûé îïåðàòîðX = t2 ∂t + tx∂x + ty∂y − tc∂c + (x − tu)∂u + (y − tv)∂v .8. Ïîêàçàòü, ÷òî óðàâíåíèÿ âðàùàþùåéñÿ ìåëêîé âîäû (g, f = const)ut + uux + vuy − f v + ghx = 0,vt + uvx + vvy + f u + ghy = 0,ht + (uh)x + (vh)y = 038äîïóñêàþò îïåðàòîðûX1 = ∂ t ,X2 = ∂x ,X3 = ∂y ,X4 = −y∂x + x∂y − v∂u + u∂v ,X5 = x∂x + y∂y + u∂u + v∂v + 2h∂h ,X6 = cos(f t)∂x − sin(f t)∂y − f sin(f t)∂u − f cos(f t)∂v ,X7 = sin(f t)∂x + cos(f t)∂y + f cos(f t)∂u − f sin(f t)∂v ,X8 = cos(f t)∂t − f2 x sin(f t) − y cos(f t) ∂x −− f2 x cos(f t) + y sin(f t) ∂y ++ f2 (u − f y) sin(f t) + (v − f x) cos(f t) ∂u −− f2 (u + f y) cos(f t) − (v + f x) sin(f t) ∂v + f h sin(f t)∂h ,X9 = sin(f t)∂t + f2 x cos(f t) + y sin(f t) ∂x −− f2 x sin(f t) − y cos(f t) ∂y −− f2 (u − f y) cos(f t) − (v − f x) sin(f t) ∂u −− f2 (u + f y) sin(f t) + (v + f x) cos(f t) ∂v − f h cos(f t)∂h .3Èíâàðèàíòû è èíâàðèàíòíûå ìíîãîîáðàçèÿ ïðåäûäóùåì ðàçäåëå ìû âèäåëè, ÷òî äèôôåðåíöèàëüíîå óðàâíåíèå ìîæåò äîïóñêàòü ñðàçó íåñêîëüêî èíôèíèòåçèìàëüíûõ îïåðàòîðîâ.

Êàæäîìó îïåðàòîðó ñîîòâåòñòâóåò ñâîÿ îäíîïàðàìåòðè÷åñêàÿ ãðóïïà Ëè ïðåîáðàçîâàíèé.Êîìïîçèöèÿ ïðåîáðàçîâàíèé êàæäîé îäíîïàðàìåòðè÷åñêîé ãðóïïû äàåò ïðåîáðàçîâàíèå òîãî æå âèäà. Îäíàêî, ÷òîåñëè ìû ñîñòàâèì ðàçëè÷íûå êîìïîçèöèè ïðåîáðàçîâàíèé, ïðèíàäëåæàùèå ðàçíûì îäíîïàðàìåòðè÷åñêèì ãðóïïàì? òàêîì ñëó÷àå âîçíèêàåò íîâûé îáúåêò: ìíîãîïàðàìåòðè÷åñêàÿ ãðóïïà Ëè ïðåîáðàçîâàíèé. Îñíîâíûì åå ñâîéñòâîìÿâëÿåòñÿ ñëåäóþùåå: â îêðåñòíîñòè òîæäåñòâåííîãî ïðåîáðàçîâàíèÿ ëþáàÿ ìíîãîïàðàìåòðè÷åñêàÿ ãðóïïà ¾ñîòêàíà¿èç îäíîïàðàìåòðè÷åñêèõ. Ò.å., ëþáîå ïðåîáðàçîâàíèå ìíîãîïàðàìåòðè÷åñêîé ãðóïïû ìîæíî ïîëó÷èòü â ðåçóëüòàòå39êîìáèíàöèè êîíå÷íîãî ÷èñëà áàçèñíûõ ïðåîáðàçîâàíèé, ïðèíàäëåæàùèõ åå îäíîïàðàìåòðè÷åñêèì ïîäãðóïïàì.

Ìûíå áóäåì óãëóáëÿòüñÿ â òåîðèþ ìíîãîïàðàìåòðè÷åñêèõ íåïðåðûâíûõ ãðóïï Ëè, ïîñêîëüêó âñþ íåîáõîäèìóþ èíôîðìàöèþ î ñòðîåíèè ýòèõ ãðóïï ìîæíî ïîëó÷èòü íà ÿçûêå àëãåáð Ëè, îáðàçóåìûõ èíôèíèòåçèìàëüíûìè îïåðàòîðàìè.3.1Àëãåáðû Ëè îïåðàòîðîâÐàññìàòðèâàåòñÿ ïðîñòðàíñòâî L ëèíåéíûõ äèôôåðåíöèàëüíûõ îïåðàòîðîâ ïåðâîãî ïîðÿäêà:L = {ξαi (x)∂xi , α = 1, .

. . , r}.Îïðåäåëåíèå 14.ôîðìóëîéÊîììóòàòîðîì îïåðàòîðîâ X = ξ i (x)∂xi è Y = η i (x)∂xi íàçûâàåòñÿ îïåðàòîð [X, Y ], îïðåäåëÿåìûékk∂η∂ξ[X, Y ] = XY − Y X = ξ i (x) i − η i (x) i ∂xk .∂x∂x(3.1)Äëÿ ëþáûõ X1 , X2 , X3 ∈ L âûïîëíåíû ñëåäóþùèå ñâîéñòâà êîììóòàòîðà.1. Áèëèíåéíîñòü: [c1 X1 + c2 X2 , X3 ] = c1 [X1 , X3 ] + c2 [X2 , X3 ], ∀ c1 , c2 ∈ R.2. Àíòèñèììåòðè÷íîñòü: [X1 , X2 ] = −[X2 , X1 ].3. Òîæäåñòâî ßêîáè: [[X1 , X2 ], X3 ] + [[X2 , X3 ], X1 ] + [[X3 , X1 ], X2 ] = 0.4.

Èíâàðèàíòíîñòü îòíîñèòåëüíî çàìåíû êîîðäèíàò: ïóñòü ïðè çàìåíå (x) ↔ (y) îïåðàòîðû ïåðåïèñûâàþòñÿ, êàêX ↔ X 0 . Òîãäà [X10 , X20 ] = [X1 , X2 ]0 .5. Èíâàðèàíòíîñòü îòíîñèòåëüíî ïðîäîëæåíèÿ: [X1 , X2 ] = [X1 , X2 ], ∀ k ∈ N.kkk6. Ñîõðàíåíèå èíâàðèàíòíîñòè ìíîãîîáðàçèÿ: åñëè ìíîãîîáðàçèåM ⊂ Rn èíâàðèàíòíî îòíîñèòåëüíî îïåðàòîðîâ X1 , X2 , òî îíî èíâàðèàíòíî è îòíîñèòåëüíî èõ êîììóòàòîðà[X1 , X2 ].40Ëèíåéíîå ïðîñòðàíñòâî îïåðàòîðîâ Lr , dim Lr = r, çàìêíóòîå îòíîñèòåëüíî îïåðàöèè êîììóòèðîâàíèÿ, íàçûâàåòñÿ àëãåáðîé Ëè îïåðàòîðîâ.Îïðåäåëåíèå 15.Èç ñâîéñòâ êîììóòàòîðà ñëåäóåò, ÷òî ëèíåéíîå ïðîñòðàíñòâî Lr îïåðàòîðîâ, äîïóñêàåìûõ ïðîèçâîëüíîé ñèñòåìîéäèôôåðåíöèàëüíûõ óðàâíåíèé, îáðàçóåò àëãåáðó Ëè. Ïóñòü 0 < r < ∞.

Òîãäà â àëãåáðå Ëè Lr ìîæíî âûáðàòüáàçèñíûå ýëåìåíòû {X1 , . . . , Xr }. Êîììóòàòîð ëþáûõ áàçèñíûõ îïåðàòîðîâ ðàñêëàäûâàåòñÿ ïî òîìó æå áàçèñó â âèäå[Xi , Xj ] = Cijk Xk , i, j, k = 1, . . . , r.Îïðåäåëåíèå 16.Ëè Lr .(3.2)Êîíñòàíòû Cijk , ôèãóðèðóþùèå â ôîðìóëå (3.2), íàçûâàþòñÿ ñòðóêòóðíûìè êîíñòàíòàìè àëãåáðûÏðè ïåðåõîäå îò îäíîãî áàçèñà â Lr ê äðóãîìó íàáîð ñòðóêòóðíûõ êîíñòàíò èçìåíÿåòñÿ ïî òåíçîðíîìó çàêîíó,ïîýòîìó íàáîð {Cijk } íàçûâàþò òàêæå ñòðóêòóðíûì òåíçîðîì àëãåáðû Ëè Lr . Ñâîéñòâà ñòðóêòóðíûõ êîíñòàíò ñëåäóþòèç ñâîéñòâ êîììóòàòîðà îïåðàòîðîâ:k,1.