Lie-Poisson reduction and reconstruction of dynamics

Marine Fontaine

Frank Adams 1,

Around $1980$, Sophus Lie found that $\mathfrak{g}^*$ the dual of a Lie algebra $\mathfrak{g}$ carries a Poisson bracket. Actually, this bracket plays an important role in Hamiltonian mechanics, it is the hidden structure behind Hamiltonian equations. However, it is not clear that Lie understood the fact that his bracket can be obtained by a Lie-Poisson reduction process.

The theory of Lie-Poisson reduction is a method whereby one reduces the dimension of the phase space of a physical system by exploiting the "symmetries". Roughly, given an Hamiltonian viewed as a smooth real-valued function defined on the phase space and the Hamiltonian equations associated to it, the reduction methods permits us to reduce the number of variables of the system.

In the Hamiltonian description, the phase space of a physical system is often expressed as $T^*G$, the cotangent bundle of a Lie group $G$ which is endowed with a canonical Poisson bracket. Using this bracket, we can write down Hamiltonian equations whose solutions are curves in $T^*G$. The reduction process states that the whole system can be "reduced" on the quotient $T^*G/G$ meaning that the "reduced solutions" will be the curves in $T^*G$ which have been passed to the quotient. Moreover, the quotient $T^*G/G$ can be regarded as $\mathfrak{g}^*$ and then, the reduced system can be expressed and solved on $\mathfrak{g}^*$. One can also reconstruct the "non-reduced" solution from the ones we found on $\mathfrak{g}^*$.

I will describe all of this in more detail and will also adapt the talk according to your questions or preferences. My aim is to show you one of many examples of the use of Lie algebras in Hamiltonian mechanics.

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