Geometric mechanics

Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory.

Geometric mechanics applies principally to systems for which the configuration space is a Lie group, or a group of diffeomorphisms, or more generally where some aspect of the configuration space has this group structure. For example, the configuration space of a rigid body such as a satellite is the group of Euclidean motions (translations and rotations in space), while the configuration space for a liquid crystal is the group of diffeomorphisms coupled with an internal state (gauge symmetry or order parameter).

One of the principal ideas of geometric mechanics is reduction, which goes back to Jacobi's elimination of the node in the 3-body problem, but in its modern form is due to K. Meyer (1973) and independently J.E. Marsden and A. Weinstein (1974), both inspired by the work of Smale (1970). Symmetry of a Hamiltonian or Lagrangian system gives rise to conserved quantities, by Noether's theorem, and these conserved quantities are the components of the momentum map J. If P is the phase space and G the symmetry group, the momentum map is a map ${\displaystyle \mathbf {J} :P\to {\mathfrak {g}}^{*}}$, and the reduced spaces are quotients of the level sets of J by the subgroup of G preserving the level set in question: for ${\displaystyle \mu \in {\mathfrak {g}}^{*}}$ one defines ${\displaystyle P_{\mu }=\mathbf {J} ^{-1}(\mu )/G_{\mu }}$, and this reduced space is a symplectic manifold if ${\displaystyle \mu }$ is a regular value of J.

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