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

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

$$M_1(f)= (x+p l)^2$$

$$M_1(M_2)= M_2 M_1= (x=x+(p+k x) l, p=p+k x) $$

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 $$x_0$$. 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
An algebra with the properties of the derivative. Related to field of non-standard analysis. TPSA vectors are approximate examples of. See also
 * differential algebra :

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).
 * TPSA:

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
 * k-Jets:

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.
 * compositional map:

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 transformation:

In general, any vector field that also has a multiplication property that satisfies In classical dynamics, refers to either phase space functions with Poisson bracket as multiplication, or Lie operators with commutation as multiplication
 * lie algebra:
 * bilinear
 * anti-commutative
 * Jacobi identity

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
 * Floquet space:

A formula relating the combining of two exponential operators into a single operator. For finite matrices, we state
 * BCH formula
 * $$ e^A e^B = e^C$$

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