On the integration of the equations of hydrodynamics

Transcription

1 Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious aticle (this jounal v 54 pp 54) I developed a theoem that educed the integation of the hydodynamical equations fo stationay motion to a system of two second-ode patial diffeential equations o to the poblem of finding a minimum to a cetain integal in which the function to be integated epesents the vis viva That was achieved by expessing the velocities in tems of two new functions that would give integals of the odinay diffeential equations that come about and which would fulfill the equation of continuity identically The extension of this pocess to the case of nonstationay motion led to vey complicated equations that did not admit any eduction to a poblem in the calculus of vaiations Since then I have found that this geneal case can also be always educed to such a poblem and indeed to the integal of a function that diffes fom the vis viva only by an additional tem The substitution that leads to that esult is essentially diffeent fom the one that was applies in the afoementioned aticle Howeve both of them have in common that they single out the detemination of the pessue fom the teatment of the est of poblem and lead to equations that epesent motions of the most geneal natue that the fluid is capable of when it is independent of extenal foces Finally they have in common that the new dependent vaiables that ae employed will define integals of the esulting system of odinay diffeential equations when they ae set equal to constants Howeve wheeas that substitution leads to two second-ode patial diffeential equations fo stationay motion in the pesent poblem the poblem comes down to thee diffeential equations two of which ae fist ode and one of which is second ode The substitution that is employed lins to the usual methods of teating hydodynamical equations In fact one odinaily maes the assumption that the expession: u dx + v dy + w dz should be a complete diffeential Howeve the u v w can always be aanged in such a way that this expession educes to a two-tem diffeential; ie to the fom: which yields the equations: dϕ + m dψ

2 Clebsch On the integation of the equations of hydodynamics u = + m x v = + m y y w = + m z z which ae just the substitutions that ae applied I ema that this has a cetain elationship to the consideation of votex motions that Helmholtz (this jounal v 55 pp 5) intoduced into the theoy Hee the velocities split into one pat that is epesented by coesponding diffeential quotients of one function and a second one that does not admit such a epesentation in the slightest Those votex motions now depend upon that second pat alone ie upon the functions m ψ If one then defines the otational velocities of a fluid paticle accoding to the equations (fomula ) that wee given thee then one will have: ξ = z y y z η = z z x ζ = y y in which the function ϕ vanishes completely ( * ) ( * ) In passing that yields the poblem of putting the expession u dx + v dy + w dz into the fom dϕ + m dψ when u v w ae any given functions As is aleady nown fom the Pfaff poblem fom the equations above m ϕ ae integals of the equations: v w w u v w dx : dy : dz = : : z y z y x Since the multiplie of the equations is if one nows one integal ψ then one can find the second one m by the pinciple of the last multiplie Howeve one will actually have: v w z y = m m z y z y etc then and ϕ will actually be a complete diffeential Howeve ϕ satisfies the diffeential equation: v w w u u v v w w u u v + + = u v + w z y z y y z z y + z y If one intoduces m ψ ϑ in place of x y z as new vaiables then one will obtain:

3 Clebsch On the integation of the equations of hydodynamics 3 I shall next tun to a geneal system of equations that exhibits popeties that ae analogous to the system of hydodynamics Suppose that one has the system of equations: () V u u u u = + u + u + + un n V u u u u = + u + u + + un n V u u u u = + u + u + + u n n n n n n n u u un () = 0 n which should be coupled with the odinay diffeential equations: (3) dx dx = u dxn = u = u n Equations () can be summaized in a symbolic fom Namely if the symbol δ implies that only the x but not t ae consideed to be vaiable then: (4) δv = u u u δ x + u + + un n o also when one sets: (5) T = u + u + + u n one will get the following: (6) δ (V T) = u u u i δ x + (u i δx u δx i ) i i ϕ = v w w u u v u + v + w z y z y ϑ v w ϑ w u ϑ u v + + z y y z z y dϑ by integation (cf Jacobi Math W v I pp 44)

4 Clebsch On the integation of the equations of hydodynamics 4 We now ema that one can always give the expession: the following fom: u δx + u δx + + u n δx n δϕ + m δϕ + + m n δϕ n so we will be led to mae the substitutions: (7) u = ϕ + n m + m + + mn u ui fo the u One can then epesent the expessions i with the help of these substitutions namely: as sums of deteminants u i ui i i = Howeve if one multiplies this expession by the deteminant (u i δx u δx i ) and then sums ove i then one will get fom nown theoems: u u i i i (u i δx u δx i ) = u + u + δ x + δ x + u + u + δ x + δ x + With consideation given to equations (3) one can employ the biefe notation fo this: dm m dϕ ϕ δϕ δ m If we now intoduce this into equation (6) then the sum: u δu = δ + mδ + δϕ will combine with the pat: ϕ δ m δϕ in the sum above to give the complete vaiation of the expession:

5 Clebsch On the integation of the equations of hydodynamics 5 ϕ + m and equation (6) will then assume the following fom: (8) δ V T m = dm dϕ δϕ δ m Howeve this equation contains only vaiations δm δϕ on the ight-hand side while n + vaiables x will be vaied on the left-hand side The expession on the left that is to be vaied must then be an abitay function of the n + aguments ϕ m t and when we once moe eliminate the symbols dm / dϕ / we can state the following theoem: Theoem Equations () () can be eplaced with the system: (9) in which: + u + u + = Π ϕ + u + u + = Π m u u un = 0 n ϕ n u = + m + m + + mn and in which Π means an abitay function of t ϕ ϕ n m m n This system contains n equations of fist ode and one of second ode Afte it has been integated the u ae themselves given by the equation above V is detemined fom the equation: (0) V = Equations (3) finally come down to the system: m m + Π dϕ () = Π m dm = Π ϕ The missing integal of the system (3) which includes one equation moe than the pesent one gives the pinciple of the last multiple

6 Clebsch On the integation of the equations of hydodynamics 6 One can add that theoem to the following one which can be veified with no futhe discussion: Theoem When V is thought of as being expessed by equation (0) equations (9) will mae the integal: assume a maximum o minimum n+ V dx dx dx n These equations include an abitay function Π Meanwhile one can assume that the integal of a system of n equations of the fist ode and one equation of second ode must include just as many abitay constants as the integal of a system of n + equations of fist ode It then seems that equations (9) into which Π entes in addition will lead to moe abitay constants than the natue of the poblem pemits This suplus of abitay constants can then have no effect on the dependent functions of the oiginal poblem; it must vanish fom the expessions fo V u u u n I will now show that in fact: One can set the function Π equal to zeo without compomising the geneality of the values of V u u u n 3 Equations () have the canonical fom which as is nown allows one to give the integals of these equations a coesponding fom and to expess them by the complete solution of a patial diffeential equation In fact one can always detemine a function (W) of t ϕ ϕ ϕ n and n constants a a a n such that: () W W W m = m = mn = n W W W α = α = αn = a a an ae the integals of equations () while the α mean new constants and one has: (3) W = Π fom which the patial diffeential equations fo W will emege when one eliminates the m fom Π with the help of the fist of equations () One can now obviously intoduce the function (W) which includes just one abitay constant into the calculations in place of the function Π in which the a α ae no longe to be egaded as constants but as functions of t x x x n when one goes fom the

7 Clebsch On the integation of the equations of hydodynamics 7 odinay diffeential equations to the patial ones At the same time one can also thin of these functions a α as dependent vaiables in the equations instead of m ϕ in which the ϕ ae also eplaced with these functions in (W) We see how the functions V u can be expessed in tems of these new dependent vaiables If one intoduces equations () into the expessions fo the u then they will go to: W W W n u = n Howeve if we let W denote the function (W) when we conside it to be a function of the t x x x n then we will obviously have: W W W W a W a a a + = and we can then (again with the help of equations ()) once moe eplace the expession above fo u with the following one: (4) u = ( ϕ + W ) a a an + α + α + + α n If we futhe ema that we also have: W W W W W a W a = a a + then the expession: Π + ϕ + W W + + m = will go to the following one: ( ϕ + W ) a a an + α + α + + α n Theefoe equation (0) will immediately assume the fom: (5) V = ( ϕ + W ) a a an n t + α α α t + t ( ϕ + W ) a a + α + α +

8 Clebsch On the integation of the equations of hydodynamics 8 We now compae equations (4) (5) with equations (7) (0) We see diectly that the function ϕ + W entes in place of ϕ while the α ente in place of m the a in place of the ϕ and finally that the function Π vanishes Now equations () obviously coespond to the following ones moeove: da (6) dα = 0 = 0 which will yield equations that ae entiely simila to equations (9) when they ae solved One then ecognizes that the educed poblem to which we have now aived diffes fom the one that was contained in Theoem only by the facts that the function Π is set to zeo and that m othe symbols ente in place of the ϕ Howeve at the same time the odinay diffeential equations will become integable and when we then evet to the pevious notation we can pose the following theoem: Theoem 3 Equations () () can be eplaced with the system: (7) in which: + u + u + + un = 0 n + u + u + + un = 0 n u u un = 0 n ϕ n u = + m + m + + mn Two of these equations ae of fist ode and one of them is of second ode Once they ae integated the u will be given by the fomula above but V will be given by the fomula: (8) V = ϕ + n m + m + + m n and equations (3) will have the equations: + ϕ n + m + m + + mn (9) m = const ϕ = const

9 Clebsch On the integation of the equations of hydodynamics 9 fo thei integals into which a last integal entes that is obtained fom equations (3) and (9) with the help of the last multiplie Thus in that way the pesent system is educed to anothe one that includes no moe abitay constants in its geneal solution than the oiginal one did and which has the popety that the integals yield the additional odinay diffeential equations with no futhe assumptions One easily infes the following theoem: Equations (7) mae the integal: Theoem 4 (n+ ) V dx dx dx n assume a maximum o a minimum in which V is thought of as being expessed by equation (8) 4 Nothing emains fo us to do but to expess the esults that we have aived at in the case of hydodynamics fo which one has n = Let U be the foce function p the pessue and let q be the density so one has the following theoem: Theoem 5 (0) Equations: p u u u u U = + u + v + w q y z p v v v v U = + u + v + w y q y z p w w w w U = + u + v + w z q y z u v w + + = 0 y z can be lined with the poblem of finding a minimum o maximum fo the integal: p U dx dy dz q in which one sets: () U p q = ψ u + v + w + m +

10 Clebsch On the integation of the equations of hydodynamics 0 and the following expessions ae tue fo the u v w: () u = + m v = + m y y w = + m z z The integals of the equations: dx = u dy = v dz = w will then be: (3) m = const ψ = const and a thid one that the theoy of the last multiplie implies The equations to which the poblem educes will then be: 0 = ϕ ϕ ϕ m m m ψ ψ ψ y y z z y y z z 0 = m + + y y z z y y z z 0 = + m + + m + + m y y y z z z It is vey easy to go fom these equations fo the stationay state to the ones that I developed in the cited place One only emas that it follows fom the consideation that was pesented in the beginning of 3 that one can choose only one of the m ψ to be completely fee of t while the othe one will be of the fom tf + F I futhe point out that equation () will become that of the vis viva in a well-nown fom when one lets m vanish which evets to the usual assumption Belin Mach 858