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0791.58038
Gotay,
Mark J.; Isenberg,
James A.
La symplectification de la science (la géométrie symplectique
aux fondements de la physique et des mathématiques). (The symplectification
of science (symplectic geometry as a basis of physics and mathematics)).
(French)
[J] Gaz.
Math., Soc. Math. Fr. 54, 59-79 (1992). [ISSN 0224-8999]
This is a well-written vulgarizing credo in the role and importance of symplectic
geometry for our present and future understanding of mathematical physics.
After a brief historical sketch of the growing impact of geometry in physics
since the time of Lagrange, the authors give a good picture of the main
differences between Riemannian geometry, which provides means for measuring
distances and angles, and symplectic geometry, which is essentially needed
to describe oriented areas. Symplectic geometry is intimately linked with
classical mechanics, but its philosophy and ideas equally apply to other
branches of physics and to various domains in pure mathematics. The remarks
which are made on the potential future role of symplectic geometry in the
whole area of mathematics sound rather speculative. The authors further
highlight the contribution of geometric quantization. An interesting discussion
in that respect concerns attempts to predict whether quantum effects could
prevent the total collapse of the universe (assuming expansion will reverse),
once its scale is reduced to a microscopic level.
[ W.Sarlet
(Gent) ]
Keywords: general expository text; symplectic geometry; mathematical physics; Riemannian geometry; geometric quantization