chain be taken so that x0 is constant; then the condition for initial values is only
dz0 − q0dy0＝0,
and the elements of the integral constituted by the characteristic chains issuing therefrom satisfy
dζ − ωdη＝0.
Hence this equation involves dz − ψdx − qdy = 0, or we have
dz − ψdx − qdy＝σ(dζ − ωdη),
where σ is not zero. Conversely, the integration of p = ψ is, essentially, the problem of writing the expression dz − ψdx − qdy in the form σ(dζ − ωdη), as must be possible (from what was said under Pfaffian Expressions).
To integrate a system of simultaneous equations of the first order X1 = a1, ... Xr = ar in n independent variables x1, ... xn and one dependent variable z, we write p1 for dz/dx1, &c., and attempt to find n + 1 − r further functions Z, Xr+1 ... Xn, such that the equations Z = a, Xi = ai,(i = 1, ... n) System of equations of the first order.involve dz − p1dx1 − ... − pndxn = 0. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xi = ai; thus each of the expressions [XiXj] must vanish in virtue of the equations expressing the integral, and we may without loss of generality assume that each of the corresponding 12r(r − 1) expressions formed from the r given differential equations vanishes in virtue of these equations. The determination of the remaining n + 1 − r functions may, as before, be made to depend on characteristic chains, which in this case, however, are manifolds of r dimensions obtained by integrating the equations [X1ƒ] = 0, ... [Xrƒ] = 0; or having obtained one integral of this system other than X1, ... Xr, say Xr+1, we may consider the system [X1ƒ] = 0, ... [Xr+1ƒ] = 0, for which, again, we have a choice; and at any stage we may use Mayer’s method and reduce the simultaneous linear equations to one equation involving parameters; while if at any stage of the process we find some but not all of the integrals of the simultaneous system, they can be used to simplify the remaining work; this can only be clearly explained in connexion with the theory of so-called function groups for which we have no space. One result arising is that the simultaneous system p1 = φ1, ... pr = φr, wherein p1, . . . pr are not involved in φ1, . . . φr, if it satisfies the 12r(r − 1) relations [pi − φi, pj − φj] = 0, has a solution z = ψ(x1, ... xn), p1 = dψ/dx1, ... pn = dψ/dxn, reducing to an arbitrary function of xr+1, ... xn only, when x1 = xº1, ... xr = xºr under certain conditions as to developability; a generalization of the theorem for linear equations. The problem of integration of this system is, as before, to put
dz − φ1dx1 − ... − φrdxr − pr+1dxr+1 − ... − pndxn
into the form σ(dζ − ωr+1 + dξr+1 − ... − ωndξn); and here ζ, ξr+1, ... ξn, ωr+1, ... ωn may be taken, as before, to be principal integrals of a certain complete system of linear equations; those, namely, determining the characteristic chains.
If L be a function of t and of the 2n quantities x1, ... xn, ẋ1, ... ẋn, where ẋi, denotes dxi/dt, &c., and if in the n equations
we put pi = dL/dẋi, and so express ẋi, ... ẋn in terms of t, xi, ... xn, p1, ... pn, assuming that the determinant of the quantities d²L/dxidẋj is not zero; if, further, H denote the function of t, x1, ... xn, p1, ... pn, numerically equal to p1ẋ1 + ... + pnẋn − L, it is easy Equations of dynamics. to prove that dpi/dt = −dH/dxi, dxi/dt = dH/dpi. These so-called canonical equations form part of those for the characteristic chains of the single partial equation dz/dt + H(t, x1, ... xn, dz/dx1, ..., dz/dxn) = 0, to which then the solution of the original equations for x1 ... xn can be reduced. It may be shown (1) that if z = ψ(t, x1, ... xn, c1, .. cn) + c be a complete integral of this equation, then pi = dψ/dxi, dψ/dci = ei are 2n equations giving the solution of the canonical equations referred to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that if xi = Xi(t, x01, ... pºn), pi = Pi(t, xº1, ... p0n) be the principal solutions of the canonical equations for t = t0, and ω denote the result of substituting these values in p1dH/dp1 + ... + pndH/dpn − H, and Ω = ∫tt0 ωdt, where, after integration, Ω is to be expressed as a function of t, x1, ... xn, xº1, ... xºn, then z = Ω + z0 is a complete integral of the partial equation.
A system of differential equations is said to allow a certain continuous group of transformations (see Groups, Theory of) when the introduction for the variables in the differential equations of the new variables given by the equations of the group leads, for all values of the Application of theory of continuous groups to formal theories.parameters of the group, to the same differential equations in the new variables. It would be interesting to verify in examples that this is the case in at least the majority of the differential equations which are known to be integrable in finite terms. We give a theorem of very general application for the case of a simultaneous complete system of linear partial homogeneous differential equations of the first order, to the solution of which the various differential equations discussed have been reduced. It will be enough to consider whether the given differential equations allow the infinitesimal transformations of the group.
It can be shown easily that sufficient conditions in order that a complete system Π1ƒ = 0 ... Πkƒ = 0, in n independent variables, should allow the infinitesimal transformation Pƒ = 0 are expressed by k equations ΠiPƒ − PΠiƒ = λi1Π1ƒ + ... + λikΠkƒ. Suppose now a complete system of n − r equations in n variables to allow a group of r infinitesimal transformations (P1f, ..., Prƒ) which has an invariant subgroup of r − 1 parameters (P1ƒ, ..., Pr−1ƒ), it being supposed that the n quantities Π1ƒ, ..., Πn-rƒ, P1ƒ, ..., Prƒ are not connected by an identical linear equation (with coefficients even depending on the independent variables). Then it can be shown that one solution of the complete system is determinable by a quadrature. For each of ΠiPσƒ − PσΠif is a linear function of Π1ƒ, ..., Πn-rƒ and the simultaneous system of independent equations Π1ƒ = 0, ... Πn-rƒ = 0, P1ƒ = 0, ... Pr−1ƒ = 0 is therefore a complete system, allowing the infinitesimal transformation Prƒ. This complete system of n − 1 equations has therefore one common solution ω, and Pr(ω) is a function of ω. By choosing ω suitably, we can then make Pr(ω) = 1. From this equation and the n − 1 equations Πiω = 0, Pσω = 0, we can determine ω by a quadrature only. Hence can be deduced a much more general result, that if the group of r parameters be integrable, the complete system can be entirety solved by quadratures; it is only necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n − r equations in n − 1 variables, subject to an integrable group of r − 1 parameters, and to continue this process. We give some examples of the application of the theorem. (1) If an equation of the first order y′ = ψ(x, y) allow the infinitesimal transformation ξdƒ/dx + ηdƒ/dy, the integral curves ω(x, y) = y0, wherein ω(x, y) is the solution of dƒ/dx + ψ(x, y) dƒ/dy = 0 reducing to y for x = x0, are interchanged among themselves by the infinitesimal transformation, or ω(x, y) can be chosen to make ξdω/dx + ηdω/dy = 1; this, with dω/dx + ψdω/dy = 0, determines ω as the integral of the complete differential (dy − ψdx)/(η − ψξ). This result itself shows that every ordinary differential equation of the first order is subject to an infinite number of infinitesimal transformations. But every infinitesimal transformation ξdƒ/dx + ηdƒ/dy can by change of variables (after integration) be brought to the form dƒ/dy, and all differential equations of the first order allowing this group can then be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of the second order y ″= ψ(x, y, y′), equivalent to dy/dx = y1, dy1/dx = ψ(x, y, y1), if H, H1 be the solutions for y and y1 chosen to reduce to yo and yo1 when x = xo, and the equations H = y, H1= y1 be equivalent to ω = yo, ω1 = yo1, then ω, ω1 are the principal solutions of Πƒ = dƒ/dx + y1dƒ/dy + ψdƒ/dy1 = 0. If the original equation allow an infinitesimal transformation whose first extended form (see Groups) is Pƒ = ξdƒ/dx + ηdƒ/dy + η1dƒ/dy1, where η1δt is the increment of dy/dx when ξδt, ηδt are the increments of x, y, and is to be expressed in terms of x, y, y1, then each of Pω and Pω1 must be functions of ω and ω1, or the partial differential equation Πƒ must allow the group Pƒ. Thus by our general theorem, if the differential equation allow a group of two parameters (and such a group is always integrable), it can be solved by quadratures, our explanation sufficing, however, only provided the form Πƒ and the two infinitesimal transformations are not linearly connected. It can be shown, from the fact that η1 is a quadratic polynomial in y1, that no differential equation of the second order can allow more than 8 really independent infinitesimal transformations, and that every homogeneous linear differential equation of the second order allows just 8, being in fact reducible to d²y/dx² = 0. Since every group of more than two parameters has subgroups of two parameters, a differential equation of the second order allowing a group of more than two parameters can, as a rule, be solved by quadratures. By transforming the group we see that if a differential equation of the second order allows a single infinitesimal transformation, it can be transformed to the form F(x, dγ/dx, d2γ/dx2); this is not the case for every differential equation of the second order. (3) For an ordinary differential equation of the third order, allowing an integrable group of three parameters whose infinitesimal transformations are not linearly connected with the partial equation to which the solution of the given ordinary equation is reducible, the similar result follows that it can be integrated by quadratures. But if the group of three parameters be simple, this result must be replaced by the statement that the integration is reducible to quadratures and that of a so-called Riccati equation of the first order, of the form dy/dx = A + By + Cy², where A, B, C are functions of x. (4) Similarly for the integration by quadratures of an ordinary equation yn = ψ(x, y, y1, ... yn−1) of any order. Moreover, the group allowed by the equation may quite well consist of extended contact transformations. An important application is to the case where the differential equation is the resolvent equation defining the group of
