# Extensions of the longitudinal envelope equation Page: 3 of 8

d2zo EL 3A
=- 3AN - K(s)zo (5)
ds2 zo 2zo
where A = q2g/(47rE0mp2c2)= rqg/p2 (rq is the classical particle radius), and K(s)= qE0'(s)/(mp2c2)
indicates an external linear bunching force. Note that, with this bunch shape and bunching force,
the forces on individual particles are linear. We may write:
d2z _3AN z_
ds2 2 - K(s)z (6)
ds2 2 z3
0
as an equation of motion for individual particles in the bunch. Note that we have dropped
relativistic factors in the motion.
Relativistic Envelope Equation
The above equations are adequate for heavy ions in linear motion. However for protons and
electrons, the motion is often somewhat relativistic, and it is desirable to change these equations
to include relativistic motion factors. Smith0 has given a (non-consistent) relativistic form of
the envelope equation; we will rederive it here in a self-consistent form.
It is somewhat tricky to obtain correct relativistic forms in all terms. To improve our chances,
we will switch to canonically-correct coordinates from the previous z-z' case, but retain position
along the accelerator, s, as our independent variable. Our variables are relative position z and
scaled momentum A, where A = Sp/(mc) = S(py), and Sp indicates the difference from the
central beam momentum. In these coordinates, the equations of motion are:
dz _ A
ds gy3 (7)
dA = -K,(s)z - q2 g dk(z)
ds 4iE0 mc2 py2 dz
where KA(s) = -q E'(s) /(m c2p) =g3 K(s).
Following the same method shown in ref. 4 (Take a wild guess and test it.), we can obtain a
relativistic envelope equation with a self-consistent distribution from the Vlasov equation. The
phase-space distribution function is:

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