Entry
Math: Calculus: Differential equation: Partial: Can you tell more about Laplace equation? [law]
Apr 16th, 2005 03:26
Knud van Eeden,
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--- Knud van Eeden --- 16 April 2005 - 05:07 pm ----------------------
Math: Calculus: Differential equation: Partial: Can you tell more
about Laplace equation? [law]
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Nature's favourite pattern?
Another striking fact is that in nature we sometimes find the same
pattern again and again in different contexts, as if the supply of
suitable patterns were extremely limited. The pattern which mathematics
denote by nabla^2(V) occurs in at least a dozen different branches of
science.
It arises in connection with
-gravitation
-light
-sound
-heat
-magnetism
-electrostatics
-electric currents
-electromagnetic radiation
-waves at sea
-the flight of aeroplanes
-vibrations of elastic bodies
-the mechanics of the atom
-- not to mention a pure mathematical theory of first class
importance, the theory of functions f(x + iy), where i is the
SquareRoot(-1).
Practical men often make the mistake of treating all these applications
as quite separate and distinct.
This is a great waste of effort.
We have not twelve theories, but one theory with twelve applications.
The same pattern appears throughout.
Physically the applications are distinct, mathematically they are
identical.
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The idea of the same pattern arising in different circumstances is a
simple one.
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One only has to invent a Greek name for this idea to have on of the
commonest terms of modern mathematics:
'isomorphic' (from 'iso' = like and 'morphe' = shape --
thus 'having the same shape').
Nothing delights a mathematician more than to discover that two
things, previously regarded as entirely distinct, are mathematically
identical.
'Mathematics', said Poincar<e'>, 'is the art of giving the
same name to different things'.
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One might ask,
'Why does this one pattern, nabla^2(V), occur again and again?'.
Here we tremble on the brink of mathematical mysticism.
There can be no final answer.
For suppose we show that this pattern has certain properties which
make it
particularly suitable; we then have to ask, 'But why does Nature
prefere those properties?' - in endless mazes lost.
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The impossibility of any final answer to the question,
'Why is the universe like it is?'
does not therefore mean that the inquiry is entirely useless.
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Nevertheless a certain answer can be given as to why nabla^2(V) so
frequently occurs.
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The explanation is along the following lines.
In empty space every point is as good as every other point, and every
direction as good as every other direction.
Laws holding in empty space may therefore be expected not to single out
any particular point or direction.
This considerably restricts the choice of possible laws.
nabla^2(V) = 0
expresses in symbols the law that the value of V at any point equals
the
average value of V on a sphere with its centre at this point.
This law treats all points and all directions alike, and is the
simplest law
that does so.
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Book: see also:
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[book: source: Sawyer, Walter Warwick - prelude to mathematics -
Pelican books - 1955 - p. 13]
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Internet: see also:
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Physics: Nature: Law: Overview: Can you give an overview of the laws
of nature?
http://www.faqts.com/knowledge_base/view.phtml/aid/35527/fid/677
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