Editing Symplectic group (section)

===Classical mechanics===
The non-compact symplectic group  {{math|Sp(2''n'', '''R''')}} comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of {{math|''n''}} particles, evolving under [[Hamiltonian mechanics|Hamilton's equations]] whose position in [[phase space]] at a given time is denoted by the vector of [[canonical coordinates]],

:<math>\mathbf{z} = (q^1, \ldots , q^n, p_1, \ldots , p_n)^\mathrm{T}.</math>

The elements of the group {{math|Sp(2''n'', '''R''')}} are, in a certain sense, [[canonical transformations]] on this vector, i.e. they preserve the form of [[Hamiltonian mechanics|Hamilton's equations]].<ref>{{harvnb|Arnold|1989}} gives an extensive mathematical overview of classical mechanics. See chapter 8 for [[symplectic manifold]]s.</ref><ref name="A&M" /> If

:<math>\mathbf{Z} = \mathbf Z(\mathbf z, t) = (Q^1, \ldots , Q^n, P_1, \ldots , P_n)^\mathrm{T}</math>

are new canonical coordinates, then, with a dot denoting time derivative,

:<math>\dot {\mathbf Z} = M({\mathbf z}, t) \dot {\mathbf z},</math>

where

:<math>M(\mathbf z, t) \in \operatorname{Sp}(2n, \mathbf R)</math>

for all {{mvar|t}} and all {{math|'''z'''}} in phase space.<ref>{{harvnb|Goldstein|1980|loc=Section 9.3}}</ref>

For the special case of a [[Riemannian manifold]], Hamilton's equations describe the [[geodesic]]s on that manifold. The coordinates <math>q^i</math> live on the underlying manifold, and the momenta <math>p_i</math> live in the [[cotangent bundle]]. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is <math>H=\tfrac{1}{2}g^{ij}(q)p_ip_j</math> where <math>g^{ij}</math> is the inverse of the [[metric tensor]] <math>g_{ij}</math> on the Riemannian manifold.<ref>Jurgen Jost, (1992) ''Riemannian Geometry and Geometric Analysis'', Springer.</ref><ref name="A&M">[[Ralph Abraham (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London {{isbn|0-8053-0102-X}}</ref> In fact, the cotangent bundle of ''any'' smooth manifold can be a given a [[symplectic manifold|symplectic structure]] in a canonical way, with the symplectic form defined as the [[exterior derivative]] of the [[tautological one-form]].<ref>{{Cite book|last=da Silva|first=Ana Cannas|url=http://link.springer.com/10.1007/978-3-540-45330-7|title=Lectures on Symplectic Geometry|date=2008|publisher=Springer Berlin Heidelberg|isbn=978-3-540-42195-5|series=Lecture Notes in Mathematics|volume=1764|location=Berlin, Heidelberg|pages=9|doi=10.1007/978-3-540-45330-7}}</ref>