Lecture 1. Reduction for Mechanical Systems with Symmetry
In this first lecture, I will discuss a selection of topics from the basic principles
of reduction theory for mechanical systems with symmetry. This is a rather large
subject and many interesting aspects h ave to be eliminated or treated only briefly.
Reduction is of two sorts, Lagrangian and Hamiltonian. In each case one has a
group of symmetries and one attempts to pass the structure at hand to an appro-
priate quotient space.
With Lagrangian reduction, the crucial object one wishes to reduce is Hamil-
ton's variational principle for the Euler-Lagrange equations. On the other hand,
with Hamiltonian reduction, the crucial objects to reduce are symplectic and Pois-
son structures. We begin by recalling Hamilton's principle.The Euler-Lagrange equations and Hamilton's principle. We learn in me-
chanics that for a Lagrangian L defined on the tangent bundle TQ of a configuration
manifold Q that the Euler-Lagrange equations for a curve q(t) E Q; namely,~ 8L _ 8L = O
dt o<t oq'
(in a coordinate chart on Q), may be regarded as a map (independent of coordi-
nates) from the second order subbundle of TTQ to T*Q. This is perhaps easiest
to see through the equivalence of the Euler-Lagrange equations to Hamilton's
principle:86 = 8 J L(q(t), q(t))dt = 0.
Here, 6 is the action function, the integral of the Lagrangian along the time
derivative of a curve in Q with fixed endpoints, regarded as a function of that
curve. Hamilton's principle states that this function 6 on the space of curves has
a critical point at a curve if and only if that curve satisfies the Euler-Lagrange
equations.Euler-Poincare reduction. Let us begin with the important, but special cases
of Euler-Poincare reduction and Lie-Poisson reduction. Following this we will com-
ment on generalizations. Let G be a Lie group (finite dimensional for simplicity)
and let G act on itself by left translation and hence, by tangent lift, on its tangent
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