basic mathematics for accelerators rende steerenberg - cern - beams department cern accelerator...
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Basic Mathematics for Accelerators
Rende Steerenberg - CERN - Beams Department
CERN Accelerator SchoolIntroduction to Accelerator Physics
31 August – 12 September 2014Prague – Czech Republic
CAS - 1 September 2014 Prague - Czech republic
2Rende Steerenberg, CERN
Marathon
However, before you start a marathon, you need a proper warming-up
This overview or review of the basic mathematics used in accelerator physics is a part of the warming up
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Contents
• What Maths are needed and Why ?
• Differential Equations
• Vector Basics
• Matrices
Rende Steerenberg, CERN
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• What Maths are needed and Why ?
• Differential Equations
• Vector Basics
• Matrices
Rende Steerenberg, CERN
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Forces in Accelerators
Rende Steerenberg, CERN
Particles in our accelerator will oscillate around the circumference of the machine under the influence of external forces
Magnetic forces in the transverse plane
Electric forces in the longitudinal plane
Mainly
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Particle Motion
Rende Steerenberg, CERN
These external forces result in oscillatory motion of the particles in our accelerator
x (hor.)
y (ver.)
S or z (long.)
Transverse Motion&
Longitudinal Motion
Decompose x and y motion
ds
x’x
dx
x s
x = hor. Displacement
x’ = dx/ds = hor. angle
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What maths & Why ?
Rende Steerenberg, CERN
Optics described using matrices (also oscillations)
Electro – Magnetic fields described by vectors
Oscillations of particle in accelerator by differential equations
x (hor.)
y (ver.)S or z (long.)
'1
01
' 1
1
2
2
x
x
BLK
x
x
QFQF QFQDQD
SFSFSF SD SD
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• What Maths are needed and Why ?
• Differential Equations
• Vector Basics
• Matrices
Rende Steerenberg, CERN
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The Pendulum
• This motion in accelerators is similar to the motion of a pendulum
• The length of the pendulum is L• It has a mass m attached to it• It moves back and forth under
the influence of gravity
Rende Steerenberg, CERN
L
M
The motion of the pendulum is described by a Second Order Differential Equation
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Simple Harmonic Motion
• Position:
• Velocity:
• Acceleration:
• Restoring force:
Rende Steerenberg, CERN
θ
M
Mg
Fr
L
(provided θ is small)
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Hill’s equation
Rende Steerenberg, CERN
Second order differential equation describing a the Simple Harmonic Motion of the pendulum
The motion of the particles in our accelerators can also be described by a 2nd order differential equation
Hill’s equationWhere: • restoring forces are magnetic fields• K(s) is related to the magnetic gradients
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Position & Velocity
Rende Steerenberg, CERN
)cos(0
txx )sin(0
txdt
dx
The differential equation that describes the transverse motion of the particles as they move around our accelerator.
The solution of this second order differential equation describes a Simple Harmonic Motion and needs to be solved for different values of K
For any system, performing simple harmonic motion, where the restoring force is proportional to the displacement, the solution for the displacement will be of the form:
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Phase Space Plot
Rende Steerenberg, CERN
Plot the velocity as a function of displacement:
)cos(0
txx
)sin(0
txdt
dx x
dt
dx
x0
x0
• It is an ellipse.• As ωt advances by 2 π it repeats itself.• This continues for (ω t + k 2π), with k=0,±1, ±2,..,..etc
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Oscillations in Accelerators
Rende Steerenberg, CERN
x = displacementx’ = angle = dx/ds
ds
x’x
dx
x s
Under the influence of the magnetic fields the particle oscillate
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Transverse Phase Space Plot
Rende Steerenberg, CERN
xφ
• φ = ωt is called the phase angle• X-axis is the horizontal or vertical position (or time in longitudinal case).• Y-axis is the horizontal or vertical phase angle (or energy in longitudinal case)
This changes slightly the Phase Space plot
X’
Position:
Angle:
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• What Maths are needed and Why ?
• Differential Equations
• Vector Basics
• Matrices
Rende Steerenberg, CERN
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Scalars & Vectors
Rende Steerenberg, CERN
Scalar: Simplest physical quantity that can be completely specified by its magnitude, a single number together with the units in which it is
measured
Age TemperatureWeight
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Scalars & Vectors
Rende Steerenberg, CERN
Vector: A quantity that requires both a magnitude (≥ 0) and a direction
in space to specify it
Velocity Force(magnetic) fields
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Vectors
Rende Steerenberg, CERN
Cartesian coordinates(x,y)
Polar coordinates(r,θ)
r is the length of the vector
22yxr
y
x
r
θ gives the direction of the vector
x
y
x
yarctan)tan(
r
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Vector Addition - Subtraction
Rende Steerenberg, CERN
Vector components
i
k
jv
addition
a
b
a + b = b + a
Subtraction
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Vector Multiplication
Rende Steerenberg, CERN
Two products are commonly defined:• Vector product vector• Scalar product just a number
A third is multiplication of a vector by a scalar:• Same direction as the original one but proportional magnitude• The scalar can be positive, negative or zero
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Scalar Product
Rende Steerenberg, CERN
The scalar product is also called dot product
a
b
θ
with
The scalar product of two vectors a and b is the magnitude vector a multiplied by the projection of vector b onto vector a
a and b are two vectors in the in a plane separated by angle θ
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Use of Scalar Products in Physics
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• Test if an angle between vectors is perpendicular
• Determine the angle between two vectors, when expressed in Cartesian form
• Find the component of a vector in the direction of another
a
b
θ
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Vector Product
Rende Steerenberg, CERN
θ
b
a
a × b
The cross product a × b is defined by:• Direction: a × b is perpendicular (normal) on the plane through a and b• The length of a × b is the surface of the parallelogram formed by a and b
a and b are two vectors in the in a plane separated by angle θ
The vector product is also called cross product
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Use of Vector Products in Physics
Rende Steerenberg, CERN
Force
B field
Direction of current
The Lorentz force is a magnetic field
The reason why our particles move around our “circular” accelerators under the influence of the magnetic fields
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• What Maths are needed and Why ?
• Differential Equations
• Vector Basics
• Matrices
Rende Steerenberg, CERN
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Matrices
Rende Steerenberg, CERN
In physics applications we often encounter sets of simultaneous linear equations. In general we may have M equations with N unknowns, of which some may be expressed by a single matrix equation.
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Addition & Subtraction
Rende Steerenberg, CERN
Adding matrices means simply adding all corresponding individual cells of both matrices an putting the result at the same cell in the sum matrix
The matrices must be of the same dimension (i.e. both M × N)
Subtraction is similar to addition e.g.: S12 = a12 + (-b12)
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Matrix multiplication
Rende Steerenberg, CERN
Multiplication by a scalar:
Multiplication of a matrix and a column vector
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Multiplication is distributive over addition:
Multiplication is not commutative:
Matrix multiplication is associative:
Matrix multiplication
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Multiplication of two matrices
and
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Null & Identity Matrix
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Null Matrix (exceptional case)
Identity Matrix (exceptional case)
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Transpose
Rende Steerenberg, CERN
The transpose of a matrix A, often written as AT is simply the matrix whose columns are the rows of matrix A
If A is an M×N matrix then AT is an N × M matrix
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Trace of a Matrix
Rende Steerenberg, CERN
Sometime one wishes to derive a single number from a matrix which is denoted by Tr A. This trace of A quantity is defined as the sum of the diagonal elements of the matrix
The trace is only defined for square matrices
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Determinant of a Matrix
Rende Steerenberg, CERN
For a given matrix the determinant Det(A) is a single number that depends upon the elements of AIf a matrix is N × N then its determinant is denoted by:
The determinant is only defined for square matrices
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Cofactor and Minor
Rende Steerenberg, CERN
In order to define the determinant of an N × N matrix we will need the cofactor and the minor
The minor of Mij of the element Aij of an N × N matrix A is the determinant of the (N-1) × (N-1) matrix obtained by removing all the elements of the ith row and jth column of A.
The associated cofactor is found by multiplying the minor by the result of (-1)i+j
Lets look at an example using:
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Cofactor of a Matrix
Rende Steerenberg, CERN
Removing all the elements of the 2nd row and the 3rd column of matrix A and forming the determinant of the remainder term gives the minor
Multiplying the minor by (-1)2+3 = (-1)5 = -1 then gives
The determinant is the sum of the products of the elements of any row or column and their cofactor (Laplace expansion).
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Determinant of a Matrix
Rende Steerenberg, CERN
As an example the first of these expansions, using the elements of the 2nd row of the determinant and their corresponding cofactors we can write the Laplace expansion
The determinant is independent of the row or column chosen
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Determinant of a Matrix
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We now need to find the order-2 determinants of the 2 × 2 minors in the Laplace expansion
Now repeat the same for the other 2 minors
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Determinant of a Matrix
Rende Steerenberg, CERN
Combing the previous:
Instead of taking the 2nd row we could have taken the first row, which would have resulted in:
Repeating this with a concrete example would result in the same scalar for both cases
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Determinant: Concrete Example
Rende Steerenberg, CERN
Use row 1:
+
+
-
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Inverse Matrix
Rende Steerenberg, CERN
If a matrix A describes the transformation of an initial vector into a final vector then one could define a matrix that performs the inverse transformation, obtaining the initial vector from the final vector. This inverse matrix is denoted as A-1
Matrix times Inverse Matrix gives the Identity matrix
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Inverse Matrix
Rende Steerenberg, CERN
If a matrix A has a determinant which is zero, then matrix A is called singular, otherwise it is non-singularIf a matrix is non-singular and then matrix A will have an inverse matrix A-1
Finding the inverse matrix A-1 can be done in several ways. One method is to construct the matrix C containing the cofactors of the elements of A.The inverse matrix can then be found by taking the transpose of C and divide by the determinant of A
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Inverse Matrix: Concrete Example
Rende Steerenberg, CERN
We previously found:
Note: this is non-zero, hence A is non-singular
Some cofactors:
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Inverse Matrix: Concrete Example
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Hence the inverse matrix of A is:
Transposing the cofactor matrix:
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Use of Inverse Matrix
Rende Steerenberg, CERN
Each element in an accelerator can be describe by a transfer matrix, describing the change of horizontal and vertical position and angle of our particle(s) • Modelling the accelerator with these transfer matrices
requires matrix multiplication• Inverse matrices will allow reconstructing initial conditions of
the beam, knowing final conditions and the transformation matrices
QFQF QF
QDQD
SFSF
SFSD SD
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Another Practical example
Rende Steerenberg, CERN
D
F
v
h
I
I
dc
ba
Q
Qor IMQ
QMI 1
• Changing the current in two sets of quadrupole magnets (F & D) changes the horizontal and vertical tunes (Qh & Qv).
• This can be expressed by the following matrix relationship:
• Change IF then ID independently and measure the changes in Qh and Qv
• Calculate the matrix M• Calculate the inverse matrix M-1
• Use now M-1 to calculate the current changes (ΔIF and ΔID) needed for any required change in tune (ΔQh and ΔQv).
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Eigenvalue & Eigenvector
Rende Steerenberg, CERN
An eigenvalue is a number that is derived from a square matrix and is usually represented by λ We say that a number is the eigenvalue for square matrix A if and only if there exists a non-zero vector x such that:
where A is a square matrix, x is the non-zero vector and λ is a non-zero value
In that case:• λ is the eigenvalue• x is the eigenvector
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Example (Normal modes)
Rende Steerenberg, CERN
Eigenvalues and eigenvectors are particularly useful in simple harmonic systems to find normal modes an displacements
m 2m
L0+x1
x x1 x2
T1L0+(x2-x1)T2 T3
5k 4k
Equations of motion for A
A B
Equations of motion for B
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Eigenvalue & Eigenvector
Rende Steerenberg, CERN
A normal mode of a mechanical system is a motion of the system in which all the masses execute simple harmonic motion with the same angular frequency called normal mode angular frequency
The equations of motion become then
In matrix form
Let: and
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Eigenvalue & Eigenvector
Rende Steerenberg, CERN
Eigenvalue problem to be solved using:
or
The normal mode angular frequencies are:
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Eigenvalue & Eigenvector
Rende Steerenberg, CERN
Now eigenvectors can be calculated to find displacement
Since the final displacements will depend on the initial conditions we can only calculate the displacement ratio between A and B
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Eigenvalue & Eigenvector
Rende Steerenberg, CERN
The example treated a simple harmonic oscillator that was coupled through springs. Using eigenvalues and eigenvectors we could conclude something about:• Oscillation frequencies• Displacements
The particles in our accelerators make simple harmonic oscillation under the influence of magnetic fields in the horizontal and vertical plane.
Through magnets, but also collective effect, the particle oscillations in the horizontal plane and vertical plane can become coupled. Eigenvalues and Eigenvector can be used to characterise this coupling
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The use of Eigenvalues & Eigenvectors
Rende Steerenberg, CERN
Under the influence of the quadrupoles the particles make oscillations that can be decomposed in horizontal and vertical oscillations:
The number of oscillations a particle makes for one turn around the accelerator is called the betatron tune:• Qh or Qx for the horizontal betatron tune• Qv or Qy for the vertical betatron tune
0 2p
x or y
Circumference
Eigenvalue and eigenvectors will provide directly information on the tunes, optics and the beam stability
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(de-)Coupling through magnets
Rende Steerenberg, CERN
Solenoid fields from the LHC Experiments cause coupling of the oscillations in the horizontal and vertical plane. This coupling needs to be compensated
Skew quadrupoles are often used to compensate for coupling introduced by magnetic errors
Eigenvalues and eigenvectors help us providing theoretical insight in this coupling phenomena
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Everything must be made as simple as possible. But not
simpler….
Rende Steerenberg, CERN
Albert Einstein
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56Rende Steerenberg, CERN
-- Spare Slides --
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57Rende Steerenberg, CERN
0)(
2
2
L
g
dt
d
)cos( tA
)sin()(
tAdt
d and )cos()( 2
2
2
tAdt
d
0)cos()cos(2
tL
gt
Differential equation describing the motion of a pendulum at small amplitudes.
Find a solution……Try a good “guess”……
Differentiate our guess (twice)
Put this and our “guess” back in the original Differential equation.
Solving a Differential Equation
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Solving a Differential Equation
Rende Steerenberg, CERN
0)cos()cos(2
tL
gt
Oscillation amplitude
tL
gA
cos
Oscillation frequency
Now we have to find the solution for the following equation:
Solving this equation gives:
The final solution of our differential equation, describing the motion of a pendulum is as we expected :