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Methods for Pseudo Invariants

This workshop presentation at the LBNL Emittance Exchange Workshop explores methods for analyzing decoupled damped systems by restoring a "determinant=1" condition through scaling the transfer map with the damping factor. The approach involves decoupling the system into 2x2 blocks and utilizing the Normal Form method for nonlinear behavior analysis. Emphasis is placed on the advantages of using pseudo-invariants to assess system quality and sensitivity to resonances, facilitating more efficient studies of dynamical behavior in both damped and non-damped coupled systems.

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Methods for Pseudo Invariants

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  1. Methods for Pseudo Invariants Decoupled Damped Systems • Restore “Determinant=1” by scaling the transfer map by the damping factor where D is the determinant of the linear map of the decoupled damped system. Coupled Systems • First, decouple the system to 2 x 2 blocks by taking real and imaginary parts of eigen vectors. The Normal Form method can be used for analysis of nonlinear behavior. M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  2. Tracking a Decoupled Damped System- Quadrupole Cooling Cells - Tracking via the damped transfer map Tracking via the determinant 1 restored transfer map of the damped system M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  3. Normal Form Theory Goal: perform a nonlinear change of variables such that the motion in the new variable pairs is rotationally invariant: If the map is symplectic, this means circles. If the map is damped, we obtain logarithmic spirals. Advantage: Tune with amplitude is trivial to compute, since each iteration of the map corresponds to the same angle advance. Other Advantages: - Provides pseudo invariants the quality of which allows conclusions about the map; - sensitive to resonances, allows efficient study of resonances. M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  4. Tracking a Coupled Non-Damped System- sFOFO 2.75m Cells from Study II - Tracking in Regular (x-x’) Coordinates 1-20 Cells 101-120 Cells Tracking in Normal Form Coordinates 1-300 Cells M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  5. Tracking a Coupled Damped System- sFOFO 2.75m Cooling Cells from Study II - Tracking in Regular (x-x’) Coordinates 1-20 Cells 101-120 Cells Tracking in Normal Form Coordinates 1-300 Cells 41-60 Cells M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

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