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< + c be a complete
integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i 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, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the
principal solutions of the canonical equations for t = t^0, and
[omega] denote the result of substituting these values in p1dH/dp1 +
... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where,
after integration, [Omega] is to be expressed as a function of t, x1,
... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral
of the partial equation.
Application of theory of continuous groups to formal theories.
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
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 [Pi]1f = 0 ... [Pi]kf = 0, in n independent variables,
should allow the infinitesimal transformation Pf = 0 are expressed by
k equations [Pi]_i Pf - P[Pi]_i f = [lambda]_i1 [Pi]1f + ... +
[lambda]_ik [Pi]_kf. Suppose now a complete system of n - r equations
in n variables to allow a group of r infinitesimal transformations
(P1f, ..., Prf) which has an invariant subgroup of r - 1 parameters
(P1f, ..., Pr-1f), it being supposed that the n quantities [Pi]1f,
..., [Pi]_n-r f, P1 f, ..., P_r f are not connected by an identical
linear equation (with coefficients even depending on the inde]]>
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