1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

388 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

Falling cat problem. This problem is an abstraction of the problem of how a

falling cat should optimally (in some sense) move its body parts so t hat it achieves

a 180° reorientation during its fall.

We begin with a Riemannian manifold Q (the configuration space of the prob-

lem) with a free and proper isometric action of a Lie group G on Q (the group

S0(3) for the falling cat). Let A denote the mechanical connection; that is, it

is the principal connection whose horizontal space is t he metric orthogonal to t he

group orbits. The quotient space Q/G = X , the shape space, inherits a Riemannian

metric from that on Q. Given a curve c(t) in Q, we sh all denote the corresponding

curve in the base space X by r ( t).

The optimal control problem under consideration is as follows:

Isoholonomic problem (falling cat problem). Fixing two points q 1 , q2 E Q,

among all curves q(t) E Q, 0 ~ t ~ 1 such that q(O) = qo, q(l) = q 1 and q(t) E

horq(t) (horizontal with respect to the mechanical connection A), find the curve or

curves q(t) sitch that the energy of the base space curve, namely,

is minimized.

~ f

1

llrll^2 dt,

2 lo

Theorem 4.1. (Montgomery [19 84 , 1990 , 1991 a]). If q(t) is a (regular) optimal

trajectory for the isoholonomic problem, then there exists a curve >.(t) E g* such that

th e reduced curve r(t) in X = Q/G together with >.(t) satisfies Wong's equations:

• 
  

  
  

  . Ba13rf3 - ~ fJgf31 Pf3P
  a a 2 fJra I
  -Aa , ca db A d ar · a
  where 9af3 is th e local representation of the metric on th e base space X ; that is

  
  

  
  

2 111 r "112_ - 2 1 ga13r ·a ·r f3 ,

gf3^1 is the inverse of th e matrix gaf3, Pa is defined by

• fJl -. (3
  Pa - fJra - ga13r ,

and where we write the components of A as A~ and similarly for its curvature B.

Proof. As with the Heisenberg system, by general principles in t he calculus of
variations, given an optimal solution q(t), t here is a Lagrange multiplier >.(t) such
t hat the new action function defined on the space of curves wit h fixed endpoints by

6[q( · )] = fo

1

[~11r(t)ll^2 + (>.(t),Aq(t))] dt

has a critical point at this curve. Using the integrand as a Lagrangian, identifying

D = Aq and applying the reduced E uler-Lagrange equations from Lecture 1 to the

reduced Lagrangian

l(r, r, D) = ~ llrll^2 + (>., D)