Routh reduction as an example of Lagrangian reduction
Presenting author: Katarzyna Grabowska
The Routh reduction is a classical piece of analytical mechanics.
It concerns systems with a special type
of symmetry, namely systems having cyclic variables. Usually the Routh reduction
is presented in terms of coordinates. The coordinate free geometric description of this
reduction is easier when tools associated with Hamiltonian mechanics
can be used. It means that we first use Legendre transformation
to pass from Lagrangian to Hamiltonian description of a system, then reduce using
symplectic geometry and then perform the Legendre transformation
backwards to get reduced Lagrangian version. Working in a coordinate free manner
we discover than the reduced Lagrangian, usually called Ruthian, is in fact not a function,
but a section of a certain one dimensional affine bundle. It is also possible to use a shortcut
i.e. reduce without passing to the Hamiltonian formalism.
It appears that Routh reduction is a very instructive example of a Lagrangian reduction,
which illustrates the principal differences between Lagrangian and Hamiltonian reductions.
The Lagrangian reduction relation, unlike the Hamiltonian one,
involves values of generating object (Lagrangians) of systems.
The talk is based on joint work with Paweł Urbański.
Click here to access the slides.