# 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:
$\epsilon =\gamma x^{2}+2\alpha xp+\beta p^{2}$
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
$A(B)=A\circ B=BA$,
For instance if
$M_{1}=(x=x+pl,p=p)\qquad M_{2}=(x=x,p=p+kx)\qquad f=x^{2}$
$M_{1}(f)=(x+pl)^{2}$
$M_{1}(M_{2})=M_{2}M_{1}=(x=x+(p+kx)l,p=p+kx)$
Please note that if A and B are matrices:
${\rm {map}}(A)\circ {\rm {map}}(B)={\rm {map}}(B){\rm {map}}(A)={\rm {map}}(AB)$
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
$g(x(t))=\exp(tf(x)\cdot \partial _{x})g(x)=\exp(v_{f})g(x)$
where $v_{f}=tf(x)\cdot \partial _{x}$ is a vector field and
$\exp(v_{f})=1+v_{f}+{\frac {1}{2}}v_{f}v_{f}+\ldots$
The method can be used for instance for solving the diff. eq. starting from an initial condition $x_{0}$. First define $g(x)=x$
then compute
$s(x,t)=\exp(v_{f})x$
then substitute $x$ with $x_{0}$ in s, and the solution will be $s(x_{0},t)$.

## 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

### 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

### 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

### BCH formula

A formula relating the combining of two exponential operators into a single operator. For finite matrices, we state
    $e^{A}e^{B}=e^{C}$
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
    $h=g+{\frac {:g:}{e^{:g:}-1}}f+...$