Vector-Valued Differential Forms
- PDF / 1,507,434 Bytes
- 52 Pages / 441 x 666 pts Page_size
- 47 Downloads / 193 Views
It is well known that the principal objects needed to define Hamiltonian dynamics are the Poisson brackets, [1, 2]. In this chapter, we discuss the different ways one can define them – through symplectic structures and more generally through Poisson structures. Naturally, here arise the questions of restriction of these structures on submanifolds, and we give some attention to this topic, since we shall use the restriction techniques heavily in the future. Finally, we discuss the questions of integrability of Hamiltonian systems and introduce in relation with the integrability questions the principal geometric object we study in this part of the book – the Nijenhuis tensor.
12.1 Symplectic Structures The classical way to define Poisson brackets, known from any course of Mechanics, is the following. Let R2n = Rnp × Rnq be the 2n dimensional Euclidean space. The notation R2n = Rnp ×Rnq means that the points x of R2n are written into the form: x = (p, q) = (p1 , p2 , . . . , pn , q 1 , q 2 , . . . , q n ) .
(12.1)
Suppose f (x), g(x) are smooth functions over R2n . Then the classical Poisson bracket {f, g} is defined as {f, g} =
n ∂f ∂g ∂g ∂f ( − ). i i ∂p ∂q ∂p i i ∂q i=1
(12.2)
The Poisson bracket of f and g is a bilinear operation and {f, g} has the properties: {f, g} = −{f, g} (skew-symmetry) {{f, g}, h} + {{g, h}, f } + {{h, f }, g} = 0 (Jacobi identity)
Gerdjikov, V.S. et al.: Hamiltonian Dynamics. Lect. Notes Phys. 748, 407–458 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-77054-1 12
408
12 Hamiltonian Dynamics
{f, gh} = {f, g}h + g{f, h}
(Leibnitz rule) .
(12.3)
In Classical Mechanics, the equations of motion of a mechanical system can be written into the so-called Hamiltonian or canonical form: p˙i =
∂H dpi =− i, dt ∂q
q˙i =
∂H dq i = ; dt ∂pi
i = 1, 2, . . . n ,
(12.4)
where H is some function on (p, q) called Hamiltonian function (or simply Hamiltonian) of the system. The variables pi are then called the generalized momenta and q i the generalized coordinates of the corresponding mechanical system. Together, (p, q) are called canonical coordinates. The above form of the equations of motion is called the canonical form of these equations. As immediately checked, the equations of motion can be cast also into the equivalent form dpi = {H, pi }, dt
dq i = {H, q i }; dt
i = 1, 2, . . . n ,
(12.5)
and moreover, if (p(t), q(t)) = x(t) is solution of the canonical equations then for each function f = f (x), we have df (x(t)) = {H, f }(x(t)) . dt
(12.6)
Then on condition that the evolution equation df = F (f (x(t))) dt
(12.7)
can be written into the form (12.6), we say that it is in Hamiltonian form with Hamiltonian function H. Thus, for writing the evolution equation for some function on R2n (the phase space), one needs only to know how to calculate Poisson brackets. This simple observation permits to make immediately important generalizations. Indeed, suppose that we have some way of defining Poisson brackets on some manifold M, that is, we can define on the space o
Data Loading...
