Math Forum Discussions

Michael Deakin

Posts: 64
Registered: 12/3/04

Re: [HM] Variation/Parameter
Posted: Feb 28, 1999 7:13 PM

Plain Text

There have been several responses to my posting on variation of
parameters, so I decided to reply to the list as a whole.

First, and sadly, I have run out of reprints of the article I cited,
so I'll give its gist below.

Lagrange did have a version of variation of parameters, but it bore
little resemblance to what we know by that term today. That
connection was made by Cauchy, and our use of the technique derives
from Cauchy.

I have few details of the anticipation by Euler. It is mentioned by
Enestrom in a footnote to Vessiot's article in the Encyclopedie
(French, not German edition). See Enc. Sci. Math. 2(16), pp. 58ff,
f/n 166. I have had no time since my last posting to add to this.
The equation discussed was

y"(x)+ky(x)=f(x)

and it arose in connection with a discussion of tides. This seems
very close to modern usage, but the Euler source is an obscure one.

I was unaware of the references to Bernoulli and Leibniz. I'll try
to follow them up -- when I get time! (A new term has just begun and
we all have massive teaching loads!)

Lagrange introduced his technique in 1774 in a discussion of
planetary orbits (Oeuvres 4, 109-147) but expanded on it in 1781
(Oeuvres 5, 123ff). For Cauchy, see Oeuvres Completes 2 ser. 6,
252-255.

Here, from a later text (Vol 3 of Price's 1856 Treatise on
Infinitesimal Calculus), is an example of Lagrange's approach, which
is essentially a perturbation technique.

Consider the equation of resisted fall

x"(t)=g-k[x'(t)]^2

and the related unperturbed equation

x"(t)=g

The latter has the solution

x=a+bt+[gt^2]/2

so for the former set

x=a(t)+b(t)t+[gt^2]/2

with

x'(t)=a'(t)+b'(t)t+b(t)+gt

In the case of the simpler equation we have

x'(t)=b+gt

and so for the perturbed equation, we expect

x'(t)=b(t)+gt.

For consistency impose the condition

a'(t)+b'(t)t=0

Putting this all together

b'(t)=-k{b(t)+gt}^2

and I leave it to you all to complete the work and reach the full
solution.

As I said, the roots of the method are present in this example (as in
Lagrange's more complicated ones) but the routine technique we use
today was extracted from it by Cauchy.

Which is not to say that Euler, Bernoulli and Leibniz didn't
also independently and earlier entertain similar thoughts!

Mike Deakin