文档介绍：Page 483
8
Applications
This chapter presents some applications of manifold theory and tensor analysis to physics and engineering.
Our selection is a of limited scope and depth, with the intention of providing an introduction to the tech-
niques. There are many other applications of the ideas of this book as well. We list below a few selected
references for further reading in the same spirit.
1. Arnol’d [1982], Abraham and Marsden [1978], Chernoff and Marsden [1974], Weinstein [1977], Marsden
[1981], Marsden [1992], and Marsden and Ratiu [1999] for Hamiltonian mechanics.
2. Marsden and Hughes [1983] for elasticity theory.
3. Flanders [1963], von Westenholz [1981], and Bloch, Ballieul, Crouch and Marsden [2001] for applications
to control theory.
4. Hermann [1980], Knowles [1981], and Schutz [1980] for diverse applications.
5. Bleecker [1981] for Yang–Mills theory.
6. Misner, Thorne, and Wheeler [1973] and Hawking and Ellis [1973] for general relativity.
Hamiltonian Mechanics
Newton’s Second Law. Our starting point is Newton’s second law in R3, which states that a particle
which has mass m>0, and is moving in a given potential field V (x) where x ∈ R3, moves along a curve
x(t) satisfying the equation of motion mx¨ = − grad V (x). If we introduce the momentum p = mx˙ and the
energy
1
H(x, p)= p2 + V (x),
2m
then the equation x˙= p/m and Newton’s law e Hamilton’s equations:
i ∂H −∂H
x˙= , p˙i = i ,i=1, 2, 3.
∂pi ∂x
484 8. Applications
To study this system of first-order equations for given H, we introduce the matrix
0 I
J = ,
−I 0
where I is the 3 × 3 identity; note that the equations e
ξ˙= J grad H(ξ)
where ξ=(x, p). plex notation, setting z = x + ip, they may be written as
∂H
z˙= −2i .
∂z¯
Suppose we make a change of coordinates, w = f(ξ), where f : R6 → R6 is smooth. If ξ(t) satisfies
Hamilton’s equations, the equations satisfied by w(t) are
˙ J J ∗
w˙= Aξ= A gradξ H(ξ)=A A gradw H(ξ(w)),
i i