For each variable listed in this section, the physical units are given between square brackets, where [1] denotes a dimensionless variable.

the following canonical variables to describe the motion of particles:

X Horizontal position \(x\) of the (closed) orbit, referred to the ideal orbit [m]. \ttitemPX Horizontal canonical momentum \(p_{x}\) of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: \(\textrm{PX}=p_{x}/p_{0}\), [1]. \ttitemY Vertical position \(y\) of the (closed) orbit, referred to the ideal orbit [m]. \ttitemPY Vertical canonical momentum \(p_{y}\) of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: \(\textrm{PY}=p_{y}/p_{0}\), [1]. \ttitemT Velocity of light times the negative time difference with respect to the reference particle: \(\textrm{T}=-ct\), [m]. A positive T means that the particle arrives ahead of the reference particle. \ttitemPT Energy error, divided by the reference momentum times the velocity of light: \(\textrm{PT}=\Delta E/p_{s}c\), [1]. This value is only non-zero when synchrotron motion is present. It describes the deviation of the particle from the orbit of a particle with the momentum error DELTAP. \ttitemDELTAP Difference between the reference momentum and the design momentum, divided by the design momentum: DELTAP = \(\Delta p/p_{0}\), [1]. This quantity is used to normalize all element strengths.

The independent variable is:

S Arc length s along the reference orbit, [m].

In the limit of fully relativistic particles (\(\gamma\gg 1\), \(v=c\), \(pc=E\)), the variables T, PT used here agree with the longitudinal variables used in \cite{TRANSPORT}. This means that T becomes the negative path length difference, while PT becomes the fractional momentum error. The reference momentum \(p_{s}\) must be constant in order to keep the system canonical.

XN The normalised horizontal displacement  \(x_{n}=Re(E_{1}^{T}\,S\,Z)\), [sqrt(m)] \ttitemPXN The normalised horizontal transverse momentum  \(p_{xn}=Im(E_{1}^{T}\,S\,Z)\), [sqrt(m)] \ttitemWX The horizontal Courant-Snyder invariant  WX = \(x_{n}^{2}+p_{xn}^{2}\), [m]. \ttitemPHIX The horizontal phase  \(\phi_{x}=-\arctan(p_{xn}/x_{n})/2\pi\), [1] \ttitemYN The normalised vertical displacement  \(y_{n}=Re(E_{2}^{T}\,S\,Z)\), [sqrt(m)] \ttitemPYN The normalised vertical transverse momentum  \(p_{yn}=Im(E_{2}^{T}\,S\,Z)\), [sqrt(m)] \ttitemWY The vertical Courant-Snyder invariant  WY = \(y_{n}^{2}+p_{yn}^{2}\), [m] \ttitemPHIY The vertical phase  \(\phi_{y}=-\arctan(p_{yn}/y_{n})/2\pi\), [1] \ttitemTN The normalised longitudinal displacement  \(t_{n}=Re(E_{3}^{T}\,S\,Z)\), [sqrt(m)] \ttitemPTN The normalised longitudinal transverse momentum  \(p_{tn}=Im(E_{3}^{T}\,S\,Z)\), [sqrt(m)] \ttitemWT The longitudinal invariant  WT = \(t_{n}^{2}+p_{tn}^{2}\), [m] \ttitemPHIT The longitudinal phase  \(\phi_{t}=-atan(p_{tn}/t_{n})/2\pi\), [1]

In the above formulas the vectors \(E_{i}\) are the three complex eigenvectors, \(Z\) is the phase space vector, and the matrix S is the “symplectic unit matrix”:

Several \madxcommands refer to linear lattice functions or optical functions.

Because \madxuses the canonical momenta (\(p_{x}\), \(p_{y}\)) instead of the slopes (\(x^{\prime}\), \(y^{\prime}\)), the definitions of the linear lattice functions differ slightly from those in Courant and Snyder\cite{Courant_Snyder1958}.

Notice that in \madx, PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respect to PT. And since PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions given by \madxmust be multiplied by BETA a number of time equal to the order of the derivative to find the functions given in the literature.

The linear lattice functions are known to \madxunder the following names:

BETX Amplitude function \(\beta_{x}\), [m]. \ttitemALFX Correlation function \(\alpha_{x}=-\frac{1}{2}(\partial\beta_{x}/\partial s)\), [1] \ttitemMUX Phase function \(\mu_{x}=\int ds/\beta_{x}\), [\(2\pi\)] \ttitemDX Dispersion of \(x\): \(D_{x}=(\partial x/\partial p_{t})\), [m] \ttitemDPX Dispersion of \(p_{x}\): \(D_{px}=(\partial p_{x}/\partial p_{t})/p_{s}\), [1] \ttitemBETY Amplitude function \(\beta_{y}\), [m] \ttitemALFY Correlation function \(\alpha_{y}=-\frac{1}{2}(\partial\beta_{y}/\partial s)\), [1] \ttitemMUY Phase function \(\mu_{y}=\int ds/\beta_{y}\), [\(2\pi\)] \ttitemDY Dispersion of \(y\): \(D_{y}=(\partial y/\partial p_{t})\), [m] \ttitemDPY Dispersion of \(p_{y}\): \(D_{py}=(\partial p_{y}/\partial p_{t})/p_{s}\), [1] \ttitemR11, R12, R21, R22 : Coupling Matrix

Several \madxcommands refer to the chromatic functions.

Because \madxuses the canonical momenta (\(p_{x}\), \(p_{y}\)) instead of the slopes (\(x^{\prime}\), \(y^{\prime}\)), the definitions of the chromatic functions differ slightly from those in \cite{Montague1979}.

Notice also that in \madx, PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respects to PT. And since PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions given by \madxmust be multiplied by BETA a number of time equal to the order of the derivative to find the functions given in the literature.

The chromatic functions are known to \madxunder the following names:

WX Chromatic amplitude function \(W_{x}=\sqrt{a_{x}^{2}+b_{x}^{2}}\) , [1], where

PHIX Chromatic phase function \(\Phi_{x}=\arctan(a_{x}/b_{x})\), [\(2\pi\)] \ttitemDMUX Chromatic derivative of phase function: \(DMUX=(\partial\mu_{x}/\partial p_{t})\), [\(2\pi\)] \ttitemDDX Chromatic derivative of dispersion \(D_{x}\) : \(DDX=\frac{1}{2}(\partial^{2}x/\partial p_{t}^{2})\), [m] \ttitemDDPX Chromatic derivative of dispersion \(D_{px}\) : \(DDPX=\frac{1}{2}(\partial^{2}p_{x}/\partial p_{t}^{2})/p_{s}\), [1] \ttitemWY Chromatic amplitude function \(W_{y}=\sqrt{a_{y}^{2}+b_{y}^{2}}\) , [1], where

PHIY Chromatic phase function \(\Phi_{y}=\arctan(a_{y}/b_{y})\), [\(2\pi\)] \ttitemDMUY Chromatic derivative of phase function: \(DMUY=(\partial\mu_{y}/\partial p_{t})\), [\(2\pi\)] \ttitemDDY Chromatic derivative of dispersion \(D_{y}\) : \(DDY=\frac{1}{2}(\partial^{2}y/\partial p_{t}^{2})\), [m] \ttitemDDPY Chromatic derivative of dispersion \(D_{py}\) : \(DDPY=\frac{1}{2}(\partial^{2}p_{y}/\partial p_{t}^{2})/p_{s}\), [1]

After a successful TWISS command a summary table, with name SUMM, is created which contains the following variables:

LENGTH The length of the machine, [m]. \ttitemORBIT5 The T (\(=ct\), [m]) component of the closed orbit. \ttitemALFA The momentum compaction factor \(\alpha_{c}\), [1]. \ttitemGAMMATR The transition energy \(\gamma_{tr}\), [1]. \ttitemQ1 The horizontal tune \(Q_{1}\) [1]. \ttitemDQ1 The horizontal chromaticity \(dq_{1}=\partial Q_{1}/\partial p_{t}\), [1] \ttitemBETXMAX The largest horizontal \(\beta_{x}\), [m]. \ttitemDXMAX The maximum of the absolute horizontal dispersion \(D_{x}\), [m]. \ttitemDXRMS The r.m.s. of the horizontal dispersion \(D_{x}\), [m]. \ttitemXCOMAX The maximum of the absolute horizontal closed orbit deviation [m]. \ttitemXRMS The r.m.s. of the horizontal closed orbit deviation [m]. \ttitemQ2 The vertical tune \(Q_{2}\) [1]. \ttitemDQ2 The vertical chromaticity \(dq_{2}=\partial Q_{2}/\partial p_{t}\), [1] \ttitemBETYMAX The largest vertical \(\beta_{y}\), [m]. \ttitemDYMAX The maximum of the absolute vertical dispersion \(D_{y}\), [m]. \ttitemDYRMS The r.m.s. of the vertical dispersion \(D_{y}\), [m]. \ttitemYCOMAX The maximum of the absolute vertical closed orbit deviation [m]. \ttitemYCORMS The r.m.s. of the vertical closed orbit deviation [m]. \ttitemDELTAP Momentum difference, divided by the reference momentum: DELTAP = \(\Delta p/p_{0}\) [1]. \ttitemSYNCH_1 First synchrotron radiation integral \ttitemSYNCH_2 Second synchrotron radiation integral \ttitemSYNCH_3 Third synchrotron radiation integral \ttitemSYNCH_4 Fourth synchrotron radiation integral \ttitemSYNCH_5 Fifth synchrotron radiation integral

Notice that in \madx, PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respects to PT. And since PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions given by \madxmust be multiplied by BETA a number of time equal to the order of the derivative to find the functions given in the literature.

The command RUN writes tables with the following variables:

X Horizontal position \(x\) of the orbit, referred to the ideal orbit [m]. \ttitemPX Horizontal canonical momentum \(p_{x}\) of the orbit referred to the ideal orbit, divided by the reference momentum. \ttitemY Vertical position \(y\) of the orbit, referred to the ideal orbit [m]. \ttitemPY Vertical canonical momentum \(p_{y}\) of the orbit referred to the ideal orbit, divided by the reference momentum. \ttitemT Velocity of light times the negative time difference with respect to the reference particle, \(\hbox{\tt T}=-c\Delta t\), [m]. A positive T means that the particle arrives ahead of the reference particle. \ttitemPT Energy difference, divided by the reference momentum times the velocity of light, [1].

When tracking Lyapunov companions, the TRACK table defines the following dependent expressions:

DISTANCE the relative Lyapunov distance between the two particles. \ttitemLYAPUNOV the estimated Lyapunov Exponent. \ttitemLOGDIST the natural logarithm of the relative distance. \ttitemLOGTURNS the natural logarithm of the turn number.