Betatron Oscillations

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The transfer matrix is now * see notes p.9 4.15 ⎡ Μ()= s a b ⎤ ⎣ ⎢ c d⎦ ⎥ ⎡ cosμ + αsinμ = βsinμ ⎣ ⎢ −γsinμ ⎤ cosμ - αsinμ⎦ ⎥ = I cosμ+Jsinμ ⎡ 4.16 J = α β ⎤ , ⎣ ⎢ −γ −α⎦ ⎥ det J =1 4.17 M Nk = ( Icosμ +Jsinμ) Nk = IcosNkμ +JsinNkμ Then if µ is real, the matrix elements of M Nk remain bounded, and the motion is stable. We note that M()= 0 I, 4.19 M −1 ()= μ M( -μ), M( μ 1 +μ 2 )= M( μ 1 )M( μ 2 ) 19 Jun 2007 Accelerators: Theory and Applications 10
The definition of µ does not depend on the point s where the matrix is defined, for, if we calculate the matrix between s 1 and s 2 +L, first by going from s 1 to s 2 , then by going from s 2 to s 2 +L, if M s 2 s 1 is the matrix connecting s 1 and s 2 , while if we first go from s 1 to s 1 +L, and then from s 1 +L to s 2 +L, Then 4.20 4.21 ( ) M( s 2 + Ls 1 )= M( s 2 )M( s 2 s 1 ) ( ) ( ) (see definition 4.6) M(s + L | s) = M s + L| s + L M s = M(s | s) M(s) 2 1 2 1 1 2 1 1 M( s 2 )= M( s 2 s 1 )M( s 1 )M −1 s 2 s 1 ( ) Thus the two matrices are related by a similarity transformation. Thus if M(s )Y = λY , M ' ' (s 2)Y =λY ( ) ′ also, where Y = M −1 s 2 s 1 Y . If det M( s 2 )−λI = 0 then det M( s 1 )− λI = 0 also, and the two matrices have the same eigenvalues, hence the same value of µ. 1 19 Jun 2007 Accelerators: Theory and Applications 11
Page 1 and 2: Physics 598ACC Accelerators: Theory
Page 3 and 4: Betatron Oscillations (see “Theor
Page 5 and 6: Wronskians: (see, for example, Mors
Page 7 and 8: Now let us define the matrix for on
Page 9: for transfer matrix on p.10 cosμ=
Page 13 and 14: For the imaginary part, ( ) ′ = 0
Page 15 and 16: Now we compare 4.28 with 4.15, and
Page 17 and 18: We can also find a constant of the
Page 19 and 20: End of Lecture 19 Jun 2007 Accelera

Betatron Oscillations

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