Guide to Non-linear Dynamics in Accelerator Physics/Definitions

phase space

Phase space refers to the space in which dynamics occurs. In order to describe dynamics with a Hamiltonian, one must specify the positions and momenta, $\vec x$ and $\vec p$. Although phase space in general may be a 2N dimensional manifold with non-trivial topology(a pendulum for example, has a position coordinate that connects back on itself). Usually, however, the phase space is $\mathbb{R}^{2N}$.

observable or function

An observable or simply function is a function from the phase space to $\mathbb{R}$. They can be represented as a multivariate polynomial or approximated by a truncated taylor series. An example of distribution is:

$\epsilon=\gamma x^2 + 2 \alpha x p + \beta p^2$

An observable or function can be composed with a map. See later.

map

A map is a function of the phase space into itself. It can be represented as a vector of observables or functions. Maps can be summed and multiplied by a scalar. A map can be a constant map:

$M: \vec z \to \vec z_0$

A map can be linear. If A is a matrix:

$M={\rm map}(A,\vec z) : \vec z \to A \vec z.$

A map can be non linear as well.

$M={\rm map}(z_1=f_1(z_1,z_2,...), z_2=f_2(z_1,z_2,...) : z_1 \to A \vec z.$

Function can be composed with maps.

If a point in the face space has coordinates (x,p), f(x,p) is an observable, M = (m1(x,p),m2(x,p)) is a map where m1,m2 are observables, composition is defined by:

f(M) = f(x = m1(x,p),p = m2(x,p)).

Composition can be extended to vector of functions and therefore with maps. Maps form an algebra with the composition operation.

We denote composition with () or $\circ$ or nothing.

If A is a map and f is a function, we denote the composition operation with

$f(A) = f \circ A = Af$

If A,B are maps, we denote the composition operation with

$A(B) = A \circ B = B A$,

For instance if

$M_1 = (x=x+p l, p=p) \qquad M_2 = (x=x, p=p+k x) \qquad f = x^2$

M₁(f) = (x + pl)²

M₁(M₂) = M₂M₁ = (x = x + (p + kx)l,p = p + kx)

Please note that if A and B are matrices:

${\rm map}(A) \circ {\rm map}(B) = {\rm map}(B) {\rm map}(A) = {\rm map}(A B)$

One may consider a tracking code as an algorithm for computing a map which is an approximation of the one turn map.

operator

An operator is a function that transform a function in a function. A map is also an operator. Operators can be generated by function like derivative operators, vector fields, lie operator. Operator can be composed to form, for instance, exponential operators.

derivative operators

A derivative operator is made of various powers of derivatives and multiplications by distributions. Examples are vector fields and lie operators.

vector field

A differential operator with the form $f(x,p)\partial_x + g(x,p)\partial_p$

dynamical system

A dynamical system can be defined by the problem of solving

$\partial_t x(t) = f(x)$

where x(t) is a trajectory in $\mathbb R^n$ and f(x) is a map.

If we are interested in finding $g(x) \circ x(t)$, where g is in general a map, the solution can be written as

$g(x(t))=\exp( t f(x) \cdot \partial_x) g(x) = \exp(v_f) g(x)$

where $v_f=t f(x) \cdot \partial_x$ is a vector field and

$\exp(v_f) = 1 + v_f + \frac12 v_f v_f + \ldots$

The method can be used for instance for solving the diff. eq. starting from an initial condition x₀. First define g(x) = x

then compute

s(x,t) = exp(v_f)x

then substitute x with x₀ in s, and the solution will be s(x₀,t).

lie operator

A special case of a vector field when the map is defined by $f=J {\rm grad} H=J \partial H$ where J is the symplectic matrix.

If H is a function of x and p.

$v_f=-\partial_p H \partial_x +\partial_x H \partial_p$.

It is often denoted in the literature as

:H:

such that

:H:g = [H,g]

Other concepts

differential algebra 

An algebra with the properties of the derivative. Related to field of non-standard analysis. TPSA vectors are approximate examples of. See also [1]

TPSA

Truncated power series algebra. Algebra of power series all truncated at a particular order. Power series may be added, multiplied. Analytic functions can be defined for them. A power series can be composed with a map. Example: epsilon(z).

k-Jets

Power series vector truncated at a particular order k. A compositional map may be represented as a K-jets if the generating map maps the origin into the origin. See also [2]

compositional map

An operator generated by a map or a function equivalent to the composition of the map with another map. A compositional map may be represented as a k-jets if the generating map maps the origin into the origin.

Lie transformation

The transformation induced by a Lie operator by exponentiating. In particular, if :f: is a Lie operator, then e^:f: is a Lie transformation. The Lie Transformations form a group, a Lie group, which is also a topological group, when defined in a more general setting.

lie algebra

In general, any vector field that also has a multiplication property that satisfies

• bilinear
• anti-commutative
• Jacobi identity

In classical dynamics, refers to either phase space functions with Poisson bracket as multiplication, or Lie operators with commutation as multiplication

Floquet space

Normalized space in which particles move in circles. Connected to Floquet's theorem which is more commonly known in solid state physics as Bloch's theorem. See also [3]

BCH formula

A formula relating the combining of two exponential operators into a single operator. For finite matrices, we state

e^Ae^B = e^C

where C is composed of sums of nested commutators of A and B. Due to the formula [:f:,:g:]=:{f,g}:, this generalizes in the case of Lie operators to the statement that

e^:f:e^:g: = e^:h:

where h is a distribution on phase space. We note, however, that this is a purely formal relationship, and may in fact break down due to lack of convergence. h may be expressed in a series in different forms depending on what is considered the expansion parameter. If both f and g are considered small, then

$h = f+g +\frac{1}{2}:f:g + \frac{1}{12}(:f:^2 g+:g:^2 f)+...$

If only g is considered small, then

$h = g + \frac{:g:}{e^{:g:}-1}f + ...$