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CHAPTER 1
Hamiltonian systems
1.1 Hamiltonian systems
This chapter and the next are a first introduction to Hamiltonian problems: more advanced material is presented later as required. A good starting point for the mathematical theory of Hamiltonian systems is the textbook by Arnold (1989). MacKay and Meiss (1987) have compiled an excellent collection of important papers in Hamiltonian dynamics. The article by Berry in this collection is particularly recommended. For an introduction to the more geometric modern approach the book by Marsden (1992) is an advisable choice.
We start by describing the class of problems we shall be concerned with and by presenting some notation. Let Ω be a domain (i.e., a nonempty, open, connected subset) in the oriented Euclidean space
of the points (
p, q) = (
p1, . . .
pd,
q1, . . .,
qd). We denote by
I an open interval of the real line
of the variable
t (time);
I may be bounded,
I = (
a, b), or unbounded,
I = (−∞,
b),
I = (
a, ∞),
I = (−∞, ∞). If
H =
H(
p, q,
t) is a sufficiently smooth real function defined in the product Ω ×
I, then the
Hamiltonian system of differential equations with Hamiltonian
H is, by definition, given by
The integer d is called the number of degrees of freedom and Ω is the phase space. The product Ω × I is the extended phase space. The exact amount of smoothness required of H will vary from place to place and will not be explicitly stated, but throughout we assume at least C2 continuity, so that the right-hand side of the system (1.1) is C1 and the standard existence and uniqueness theorems apply to the corresponding initial value problem. Sometimes, the symbol SH will be used to refer to the system (1.1).
Usually, in applications to mechanics (Arnold (1989)), the q variables are generalized coordinates, the p variables the conjugated generalized momenta and H corresponds to the total mechanical energy.
In many Hamiltonian systems of interest, the Hamiltonian
H does not explicitly depend on
t; then
(1.1) is an
autonomous system of differential equations. For autonomous problems we shall consider
H as a function defined in the phase space Ω, rather than as a function defined in
and independent of the last variable.
It is sometimes useful to combine all the dependent variables in (1.1) in a 2d-dimensional vector y = (p, q). Then (1.1) takes the simple form
where ∇ is the gradient operator
and J is the 2d × 2d skew-symmetric matrix
(I and 0 respectively represent the unit and zero d × d matrices).
Upon differentiation of H with respect to t along a solution of (1.1), we find
so that, in view of (1.2) and of the skew-symmetry of J−1,
In particular, if H is autonomous, dH/dt = 0. Then H is a conserved quantity that remains constant along solutions of the system. In the applications, this usually corresponds to conservation of energy.
We now turn to some concrete examples of Hamiltonian systems. These examples have been chosen for their simplicity. More realistic examples from celestial mechanics, plasma physics, molecular dynamics etc. can be found in the literature of the corresponding fields.
1.2 Examples of Hamiltonian systems
1.2.1 The harmonic oscillator
This is the well-known system with d = 1 (one degree of freedom)
Here, m and k are positive constants that, for the familiar case of a material point attached to a spring, respectively correspond to mass and spring constant. Of course...
