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16 J. Butterﬁeld motion. This will mean that the group of symmetries deﬁning the quotienting procedure will be the rotation group. It will also mean that the rigid body’s conﬁguration space is given by the rotation group, since any conﬁguration can be labelled by the rotation that obtains it from some reference-conﬁguration. So in this application of symplectic reduction, the symmetry group (viz. the rotation group) will act on itself as the conﬁguration space. This example will also give us our prototype example of a Poisson manifold.

In particular, it induces an anti-symmetric bilinear form that may be degenerate, as in eq. 3. A manifold M equipped with such a bracket is called a Poisson manifold. 1 that any symplectic manifold is evendimensional. 1. 7 7 Because of these clear connections, it is natural to still call the more general framework ‘Hamiltonian’; as is usually done. But of course this is just a verbal matter. On Symplectic Reduction in Classical Mechanics 15 (2): Historical roots: The theory of symplectic reduction has deep historical roots in the work of classical mechanics’ monumental ﬁgures.

And much of what follows — in spirit, and even in letter — is already in Lie [1890]! As a (non-philosophical) introduction to symplectic reduction, I also recommend Singer (2001). g. it omits Poisson manifolds and co-adjoint representations. 18 J. Butterﬁeld gets by quotienting the orthodox absolutist (also called a ‘substantivalist’) classical mechanics, by an appropriate symmetry group. 10 Belot’s overall idea is as follows. Where the relationist admits one possible conﬁguration, as (roughly) a speciﬁcation of all the distances (and thereby angles) between all the parts of matter, the absolutist (or substantivalist) sees an inﬁnity of possibilities: one for each way the relationist’s conﬁguration (a relative conﬁguration) can be embedded in the absolute space.