 About 6,600 results  books.google.com The trick is to reduce the general theorem to Liouville's principle on doubly
periodic functions. To this end, Liouville composed f with Jacobi's elliptic function
sinam z, which is the same as Abel's function p(z), (cf. (1) and (2)) for c = 1. Then,
the ... 

 books.google.com From this he derived Liouville's “principle” and, significantly, admitted that in his
previous paper (1844g) he had limited himself to stress the analogy between his
own and Liouville's principle because he “did not remember of having published
... 

 books.google.com 11.3 Liouville on His Theorem for Doubly Periodic Functions 119 lectures in 1880
, by which time he had become the editor of the Journal für die reine und
angewandte Mathematik.4 Cauchy recognised the generality of the principle at
once, ... 

 books.google.com (ii) Integrate these principal parts termwise, except the terms with exponents 1, (
iii) Interpolate, i.e. find a function v e C(x) ... While this is not necessarily true for
other classes of functions (e.g. f e” dr), Liouville's principle informally states that if
a ... 

 books.google.com Proof: Suppose that f has an elementary integral F. Then by Liouville's principle
and the fact that D(e$2) I (2:62 A 22:, we see that I r(:c) ~ 6102. Therefore F'(a:) I r'
(:v)e3E2 + r(m)  23: e”2 I ex2 Dividing by e52 gives the differential equation 1"' +
... 

 books.google.com Gibbs called the Liouville condition the principle of density in phase. Following
this principle, classical statistical mechanics becomes a theory in which the
motion of particles (or systems) is deterministic, but unpredictable individually:
the ... 

 books.google.com LIOUVILLE'S CONTRIBUTIONS Liouville developed these ideas in at least three
important directions: 1° He discovered [Liouville 1855] what is ... Like Liouville's
theorem Liouville's analysis of the principle of least action was part of the content
... 

 books.google.com Indeed, it was the French mathematician Joseph Liouville who first extensively
studied the problem of integration in closed form. Liouville's principle, presented
in 1833, describes the structure of all elementary functions” whose antiderivatives
... 

 books.google.com Defining q = (q, , . . . , qN), p = (p, , . . . , p^), if o (q, p, t) satisfies Liouville's
equation then so does o (q, —p, —t). ... mechanics is timesymmetric, since
statistical mechanics is based not only on Liouville's equation but also on Jaynes'
principle. 

 books.google.com Another way to describe Liouville's principle is provided in [35]: If f{x,yi,V2, □ □ ,
ym), y[, y'2, □□□,y'm are algebraic in x,yi, ...,ym then / f{x,y1,y2,...,ym)dx is
elementary if and only if /m f{x, 2/i, 2/2, • • • , 2/m) dx = U0 + ^ C3 ln U3 3 = 1
where the ... 

 