Serge AndrianovSerge Andrianov

Theory of SymplecticTheory of Symplectic Formalism for Formalism for Spin-Orbit TrackingSpin-Orbit Tracking

Institute for Nuclear Physics Forschungszentrum JuelichInstitute for Nuclear Physics Forschungszentrum Juelich

Saint-Petersburg State University, Russia Saint-Petersburg State University, Russia

September 25, 2013September 25, 20131/121/12

Theoretical preparation of motion equations

Transport matrix presentation of solution

B and E-fields expansion in the neighborhood of the equilibrium trajectory

Lorentz-Maxwell equationsEquation of particles and spin

evolution:Particles and Spin Motion Equations

Tensor presentation of solution

Transformation of the L-M+T-BMT equations into the auxiliary coordinate system

+ T-BMT equationsL-M+T-BMT equations

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The matrix form of nonlinear ODE’s

k-times

Here the Kronecker product.

For k=2 we have

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We use Taylor series expansion in this form

All equations of particles and spin evolution can be written the following short form:

The solution is constructed in the form

The matrix presentation of ODE’s solutions (continue)

Matrices describe the beam evaluation, which have evaluate using some different schemes. The choice of the scheme defines the convergence velocity corresponding series. Here there some different possibilities!

The form of “usual” tensor presentation of the corresponding series can be written in the more complicated form

see, for example, E´tienne Forest Geometric integration for particle accelerators. J. Phys. A: Math. Gen. 39 (2006) 5321–5377

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From a physical model to a mathematical modelsIt is well known that the computational problems in beam physics can be divided into the following groups.

The First group of the problem is related to determining the required degree of approximation corresponding to the problem under study (on what power degree we should truncate our series (the accuracy of all our manipulations).

The Second group is based on calculation of the corresponding evolution operators in the framework of an used formalisms (here can be used different kind of methods – for example, COSY Infinity).

The Third group of problems is connected with some specific problems, for example - long beam evolution problem, spin dynamics, the influence space charge, different aberrations correction and so on.

It is necessary also to mention problems embedding the selected numerical methods into some computational framework to carry out the necessary computational experiments, carry out optimization process and so on.

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From a physical model to a mathematical modelsLet list the basic requirements for the methods that can be used to model the beam dynamics and the corresponding related problems.

First, the accuracy of approximation of the “ideal mapping” generated by the dynamical system under study. Here we should mention the problem: how estimate the closeness of ideal solutions and the corresponding approximate solutions?

The second important demand is connected with the need to preserve the qualitative properties inherent in the dynamical system under study. For example, such as the symplectic property for Hamiltonian systems, conservation of exact (for example, energy conservation) and approximate integrals of motion and so on.

Finally, accurate construction of the maps for some practical classes of dynamical systems. All our series will converge absolutely!

In particular, it is necessary to mention one more problem - the problem of parallel and distributed computing processes using the corresponding maps both for the dynamics of particles and for an ensemble of particles in whole. The suggested method admits distributing and paralleling by a naturally way. 6/126/12

Some examples of exact solutions

The correctness of the matrix formalism can be tested for some simple examples.

1. One-dimensional nonlinear equation

For Lie operator for our differential equation can be written in the form

The exact solution of this equality has the form

After some simple calculations one can obtain the desired solution!

!

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As an example we consider the second order nonlinear Hamiltonian equations

Using the above described approach one can obtain

where

We should note that (5) is exact solution of the equation system (4)!

(5)

(4)

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Some examples of exact solutions

The preservation of qualitative properties in matrix formalism(symplecticity)

The symplecticity property of our computational scheme constrains some special restrictions, which can be written in the linear algebraic equations for some elements of corresponding matrices. For example, for the second order we obtain

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Remark2. For two dimensional phase vector and the second order of nonlinearity have the full agreement with the result, published in the paper by E´tienne Forest “Geometric integration for particle accelerators”. J. Phys. A: Math. Gen. 39 (2006) on the page 5335:

It should be noted that similar matrices can be precomputed (for example, using Maple or Mathematica packages) and kept that in a special database. This approach guaranties us fulfillment of the symplecticity conditions up the necessary order (for arbitrary interval of independent variable)!

The symplecticity condition

Remark1. Similar formulae we can receive from linear algebraic equations and for any order of nonlinearities!

The preservation of qualitative properties in matrix formalism(energy conservation)

It is known that in general cases the symplecticity of the map (exact or approximate map) does not guarantee the energy conservation. That is why we should imposed additionally condition for our approximating maps. In another words on the every step of the integration process we should guarantee the fulfillment of energy conservation law, which can be written in the following forms.

In another words on the every step we must guarantee the energy conservation low, which can be written in the following forms

,

For linear case we have

These conditions can be realized using some correction procedure.

where , and we have !

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The preservation of qualitative properties in matrix formalism(energy conservation for nonlinear systems)

If we want to conserve nonlinear Hamiltonian, than we should “correct a little” our

truncated matrix map. In another words, some elements of we

should be corrected. For this purpose we can evaluate some equations (see, an example, the correction procedure for symplectification). Here there are some different approaches. The choice of appropriate variant depends on the practical problem: the symplectification condition is universal property, while the energy conservation depends on the energy function

(Hamiltonian)!Here there are some approaches. For our problems (particles beam + spin dynamics) the most appropriate method is the method of stroboscopic averaging, using the ergodic properties of our dynamical system.

There is a problem: Can we construct an integration scheme that is both symplectic and energy-conserving properties for a broad class of Hamiltonian systems? The well known Zhang and Marsden theorem answer – in general case – NO!

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The convergence problem for the matrix formalism (in terms of the phase vector )

We can derive corresponding conditions for convergence of matrix formalism using ODE’s presentation. Let cite corresponding estimations.

Let be , from where and we have

and , We can show that

there are the next inequalities and

Let be then we have (here is an exact

solution):

This inequality allows us to provide necessary estimation of our calculation accuracy!

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Thank for you attention!Thank for you attention!

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Saint-Petersburg State University, Russia Saint-Petersburg State University, Russia

Institute for Nuclear Physics Forschungszentrum JuelichInstitute for Nuclear Physics Forschungszentrum Juelich September 25, 2013September 25, 2013