physics 598acc accelerators: theory and applicationsweb.hep.uiuc.edu/home/derrede/598acc/lecture 1...

11 Jun 2007 Accelerators: Theory and Applications

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Physics 598ACC Accelerators: Theory and Applications

Instructors: Fred Mills, Deborah Errede

Lecture 1: Review of Classical Mechanics


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SummaryA. Lagrangians, Equations of Motion, Variational

principles applied to HamiltoniansB. E-M forces added to particle HamiltonianC. Canonical Transformations ( used often in this course )D. Adiabatic Processes


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` Review of Classical MechanicsA.Variational Principles

We will be interested in the orbits of particles in Electromagnetic (E-M) Fields. Usually, the particles will be relativistic. We want to apply all available methods to the solution of problems. The solution of mechanics problems corresponds to finding the integrals, or constants of motion, of the equations describing the motion. This is generally facilitated by choosing the appropriate coordinate system.

Newton's laws, and d'Alembert's principle, lead to the conservation of energy, and thence to the Lagrangian

1.1

where T and V are the kinetic and potential energies, respectively, and (.) means total differentiation with respect to time. The equations of motion are:

1.2

( , , )L r r t T V= −&

ii

Lpq

∂=

∂&


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where qi and pi are respectively coordinate and momentum corresponding to the "i"thdegree of freedom. pi is given by

1.3 ii

Lpq

∂=

∂ &

Aside: Lagrangian versus Hamiltonian formulation, Canonical variables Goldstein 2nd Edition(1980) p.339

“In the Lagrangian formulation (nonrelativistic) a system with n degrees of freedom possesses n equations of motion of the form

As the equations are of second order, the motion of the system is determined for all time only when 2n initial conditions are specified, e.g., the n qi’s and n ’s at a particular time t1, or the n qi’s at two times t1 and t2. We represent the state of the system by a point in an n-dimensional configuration space whose coordinates are the n generalized coordinates qi and follow the motion of the system point in time as it traverses its path in configuration space. Physically, in the Lagrangian viewpoint a system with n independent degrees of freedom is a problem in n independent variables qi(t), and appears only as a shorthand for the time derivative of qi.

i i

d L L 0dt q q

⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠&

iq&

iq&


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The Hamiltonian formulation is based on a fundamentally different picture. We seek to describe the motion in terms of first-order equations of motion. Since the number of initial conditions determining the motion must of course still be 2n, there must be 2n independent first order equations expressed in terms of 2n independent variables.Hence the 2n equations of the motion describe the behavior of the system point in a phase space whose coordinates are the 2n independent variables. In thus doubling our set of independent quantities, it is natural (though not inevitable) to choose half of them to be the n generalized coordinates qi. As we shall see, the formulation is nearly symmetric if we choose the other half of the set to be the generalized or conjugate momenta pi already introduced by the definition

The quantities (q,p) are known as the canonical variables.”

Goldstein goes on to say that from a strictly mathematical viewpoint, the transition from Lagrangian to Hamiltonian formulation corresponds to switching variables from (q,q’,t) to (q,p,t), where p is related to q and by the above equation, and that one uses the Legendre transformation to do this.

j ji

i

L(q ,q , t)p

q∂

=∂

&

&

iq&


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Legendre Transformations: Goldstein p.340let f be a function of two variables f(x,y) so that a differential has the form

where

We want to change the basis of description from x,y to a new set of variables u,y, so that the differential quantities are expressed in terms of du and dy. Let g be a function of u and y defined by the equation

then the differential becomes

or

This is the form desired. The quantities x and v are now functions of the variables u and y given by the relations

-return to lecture

fux

∂=

∂fvy

∂=

∂

g f ux= −

dg df udx xdu= − −

gxu

∂= −

∂gvy

∂=

∂

dg vdy xdu= −

df udx vdy= +

11 Jun 2007 7

If L is independent of qi, then pi is a constant of the motion. Further, the function H, defined as

1.4 Legendre transformation

satisfies

1.5

so that if L or H is independent of t, H is a constant of the motion. H is called the Hamiltonian function. If H is considered to be a function of the p's and q's, then the equations of motion become

1.6

1.7 *see notes

1.8

Then -H appears as a momentum conjugate to t.

i ii

H p q L= −∑ &

dH H Ldt t t

∂ ∂= = −

∂ ∂

ii

Hpq

∂= −

∂&

ii

Hqp

∂=

∂&

HHt

∂=

∂&


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Let’s take a look at the derivation of . Goldstein p.341

The terms cancel and using Lagrange’s equation we will get

which proves the result.

ii

Hqp

∂=

∂&

i ii i

H H HdH dq dp dtq p t

∂ ∂ ∂= + +

∂ ∂ ∂

i iH(q,p, t) q p L(q,q, t)= −& &

i i i i i ii i

L L LdH q dp p dq dq dq dtq q t

∂ ∂ ∂= + − − −

∂ ∂ ∂& & &

&

idq&

i i i iLdH q dp p dq dtt

∂= − −

∂& &

i i

d L L 0dt q q

⎛ ⎞∂ ∂− =⎜ ⎟∂ ∂⎝ ⎠&


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These equations can be derived from a variational principle, called Hamilton's Principle, δI = 0, where

1.9 *see notes

The variations in the p's and q's are = 0 at the end point times t1 and t2. We can extend Hamilton's Principle and allow variations in t and H by introducing a parameter λ as an independent variable, and minimizing I subject to the constraint F(Htpq) = 0, that is, that we know the functional form of the Hamiltonian (solve F(Htpq) = 0 for H). Here K is a Lagrange multiplier. Then if

1.10

1.11

where δ means variations of (Htpq) which are = 0 at the end points λ1 and λ2.

( )2 2

1 1

t t

i iit t

I Ldt p dq Hdt⎛ ⎞= = −⎜ ⎟⎝ ⎠∑∫ ∫

dxxdλ

′ ≡

2

1

( ) ( )i ii

p q Ht K F Htpq dλ

λ

δ λ λ⎛ ⎞′ ′− +⎜ ⎟⎝ ⎠∑∫


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Variational Principles Goldstein p. 37

where

Consider only variations in paths for which y(x1)=y1 and y(x2)=y2.

(drawing) αβχδεφγηιϕκλμνοπθρστυϖωξψζ

Let η(x) be a well-behaved function that vanishes at the endpoints. y(x,0) is the correct path.

For any such parametric family of curves J is also a function of α :

2

1

x

x

J f (y, y, x)dx= ∫ &dyydx

≡&

y(x, ) y(x,0) (x)α = + αη

2

1

x

x

J( ) f (y(x, ), y(x, ), x)dxα = α α∫ &


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Variational Principles Goldstein

and the condition for obtaining a stationary point is the familiar one that

Differentiating under the integral sign

Look at the second integral.Integrate by parts

eqn. A.1

0

dJ 0d α=

⎛ ⎞ =⎜ ⎟α⎝ ⎠

2

1

x

x

dJ f y f y dxd y y

⎧ ⎫∂ ∂ ∂ ∂= +⎨ ⎬α ∂ ∂α ∂ ∂α⎩ ⎭

∫&

&2 2

1 1

x x 2

x x

f y f ydx dxy y x

⎧ ⎫⎧ ⎫∂ ∂ ∂ ∂=⎨ ⎬ ⎨ ⎬∂ ∂α ∂ ∂ ∂α⎩ ⎭ ⎩ ⎭

∫ ∫&

& &

2 22

1 11

x xx

x xx

f y f y d f ydx dxy y dx y

⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂= −⎨ ⎬ ⎜ ⎟∂ ∂α ∂ ∂α ∂ ∂α⎩ ⎭ ⎝ ⎠

∫ ∫&

& & &


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Variational Principles Goldstein

Since all the variations in the paths at the endpoints must vanish the first derivative of y(x,α) must vanish there also.

therefore the first term in eqn A.1 vanishes.

We get

The condition for a stationary value is equivalent to

The partial derivative of y wrt α is arbitrary except for continuity and end point conditions requires that the function in brackets be zero everywhere. (the form of Lagrange’s equation ) back to lecture page 9.

1 2y(x , ) y(x , ) 0∂ α ∂ α= =

∂α ∂α

0

dJ 0d α=

⎛ ⎞ =⎜ ⎟α⎝ ⎠

2

1

x

x

dJ f d f y dxd y dx y

⎧ ⎫∂ ∂ ∂= −⎨ ⎬α ∂ ∂ ∂α⎩ ⎭

∫ &

2

1

x

x

f d f y dx 0y dx y

⎧ ⎫∂ ∂ ∂− =⎨ ⎬∂ ∂ ∂α⎩ ⎭

∫ &


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Now if we choose

1.12

where H(pqt) is the known functional form, the variation leads to the equations

1.13 ,

1.14 ,

Then K determines the relationship of λ to time. We are at liberty to choose K=1; then

1.15

subject to H = H(pqt), and the variations of (Htpq) are = 0 at t1 and t2.

HH Kt

∂′− = −∂

ii

Hq Kp

∂′ =∂i

i

Hp Kq

∂′ = −∂

( )F H pqt H= −

dtt K( )d

′ = λ =λ

2

1

t

i iit

p dq Hdt 0⎛ ⎞δ − =⎜ ⎟⎝ ⎠∑∫


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Here we have four coordinates and four momenta, and we are at liberty to choose the independent variable. The choice of t is not always the best for calculating orbits. We can choose, say, q1 to be the independent variable, and

1.16 *see notes

where now (') means differentiation with respect to q1 and the sum is over 2 and 3. now plays the role of the Hamiltonian, where we have solved for

p1(q2,p2,....,H,t). The equations of motion are

1.17 , j = 2, 3

1.18 ,

1.19 compare with eqn 1.5

If G is independent of q1, then G is a constant of the motion.

jj

Gqp

∂′ =∂ j

j

Gpq

∂′ = −∂

GtH

∂′ = −∂

GHt

∂′ =∂

1 1j jj

p p q Ht dqδ⎛ ⎞

′ ′+ −⎜ ⎟⎝ ⎠

∑∫

1 1

G dGGq dq

∂′ = =∂

1p G− ≡


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The six dimensional space comprising the ranges of the coordinates and momenta (pi,qi) is called "phase space". Liouville's theorem states that the density of points in phase space remains constant as the motion takes place. If the shape of the volume becomes contorted and filamentary, then the effective volume increases. This is called "dilution" and a major goal of accelerator design is to avoid, or at least minimize, dilution. Otherwise, either beam is lost, or large apertures must be used.

* scale transformations can be dropped directly into equation 1.16 and still satisfy Hamilton’s principle; where p -> p/C and q-> Cq, where C is a constant in the problem under consideration.


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B. E-M Forces

For non-relativistic motion, the LaGrangian is given by

1.20

φ and Α are the scalar and vector potentials of the E-M fields. The canonical momentaare

1.21

The canonical momentum is no longer the mechanical momentum, but now includes the potentials as well. The Hamiltonian is

1.22 *see Goldstein

2v eL=m e + v A2 c

− φ ⋅

i i ii

L ep mv Aq c

∂= = +

∂&

2

i ii

i ii

ep AcH p v L e

2m

⎡ ⎤−⎢ ⎥⎣ ⎦= − = + φ∑

∑


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The introduction of E-M fields corresponds to the replacement

1.23 ,

in the free space Hamiltonian

1.24

ep p Ac

→ − H H e→ − φ

2i

i

pH2m

=∑


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C. Relativistic MotionWhat is essentially different about relativistic motion is the relationship

between energy and momentum. The free particle has

1.25

With E-M fields we can use the replacement 1.23 above to obtain

1.26

We can use the same variational principles as above to obtain the equations of motion, choice of independent variable, constants of the motion, etc.

2 2 2i

iH p m c= +∑

22 2

i ii

eH p A m c ec

⎡ ⎤= − + + φ⎢ ⎥⎣ ⎦∑


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D. Canonical Transformations We would like to make general time dependent transformations between old

and new variables (pq)->(PQ)

1.27 ,

and require that the system be still Hamiltonian, that is, that there exist K such that

1.28 , ,

Then it must be true that

1.29

This will still be true if the two integrands differ by the total time derivative of some function F(pqPQt). (This is called a generating function). The transformation equations can be used to eliminate 2n of the variables, so that F only depends upon 2n of the (PQpq) and also t. There are four common choices, leading to four sets of transformation equations. Note: Because dF/dt is integrated wrt t and F(PQpq) is evaluated at the endpoints, where the variations of the canonical coordinates are zero, this added term does not contribute to the action integral.

i iQ Q (qpt)= j jP P (qpt)=

ii

KQP

∂=

∂&

jj

KPQ

∂= −

∂& KK

t∂

=∂

&

i i i ip dq Hdt 0 PdQ Kdt 0⎡ ⎤ ⎡ ⎤δ − = ⇔ δ − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑∫ ∫


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1 1 11 i j

i j

F F FF (qQt) : p , P ,K Hq Q t

∂ ∂ ∂= = − = +

∂ ∂ ∂

3 3 33 i j

i j

F F FF (Qpt) : P ,q , K HQ p t

∂ ∂ ∂= = = −

∂ ∂ ∂

4 4 44 i j

i j

F F FF (pPt) : q ,Q ,K Hp P t

∂ ∂ ∂= − = = +

∂ ∂ ∂

2 2 22 i j

i j

F F FF (qPt) : p ,Q , K Hq P t

∂ ∂ ∂= = = +

∂ ∂ ∂

some useful transformation equations to new sets of canonical variables:

1.30

1.31

1.32

1.33

We can specify a transformation by giving the function F, or by specifying the properties of the (PQ) and solving for the function F (Hamilton-Jacobi approach).


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Alternatively we can write down the transformation equations 1.27. In this case we must verify that the transformation is canonical. It can be shown that the "Poisson Brackets" (here defined) of the transformation, if canonical, satisfy

1.34

1.35

Poisson Brackets have other applications. For example, any general function f(pq) has the total time derivative

1.36

Then if {f,H}=0, f is a constant of the motion.

{ } j ji ii j i, j

k,l l k k l

Q QP PP ,Qq p p q

∂ ∂⎡ ⎤∂ ∂= − = δ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

∑

{ } { }i j i jP ,P Q ,Q 0= =

{ }df f , Hdt

=

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E. Adiabatic Processes

The idea of an adiabatic invariant Ji, a property of an orbit in an oscillating system, was introduced into mechanics by Paul Ehrenfest. He showed that these dynamical quantities in an oscillating system are adiabatic invariants in the sense that if a parameter λ in a system undergoes a change, the variation in the Ji depends in lowest order on . These quantities were used as candidates for quantization in the Bohr-Sommerfeld theory of the Hydrogen atom.

In accelerators, we often must change the value of the parameters determining the focusing strength for a particle, for example the RF voltage accelerating the particle. We usually try to do this "adiabatically" in an attempt to reduce the phase space dilution which might attend this process. K. R. Symon, using computational methods, evolved a rule for adiabatic processes in an oscillating system in which the fractional change in the parameter λ is related to the elapsed phase in the oscillation of frequency ω;

1.37

where ω is the instantaneous value during the change dλ. The constant A was found by computation, for a process not near fixed points or other singularities, by requiring that the total change be accomplished during one quarter of an oscillation or longer.

2λ&

d A ( )dtλ= ± ω λ

λ


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This led to the well-known rule for changing RF voltage;

1.38

It is well known experimentally that such transitions are extremely, even surprisingly, successful at avoiding dilution.

Consider a harmonic oscillator with mass m and spring constant k. The Hamiltonian of the system is

1.39 ,

k and m, and therefore ω and β are known functions of time.

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V(0)V(t)(1 A t)

=− ω

2 2 22p kq qH p ;

2m 2 2⎡ ⎤ω

= + = β +⎢ ⎥β⎣ ⎦

km

ω =1

mkβ =


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Although there is no longer a constant energy integral, we might expect that since this is a linear system it should be possible to obtain analytic solutions. To obtain them, let us transform to new coordinates I and γ, where

1.40

using the generating function

1.41

[Notice ]

The inverse transformation is

1.42

22 qp

I ,2

⎡ ⎤β +⎢ ⎥β⎣ ⎦=

1 ptanq

− βγ = −

2

1q tanF

2γ

= −β

q 2I cos ,= β γ 2Ip sin= − γβ

H I= ω


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and the new Hamiltonian is

1.43 *see notes

If , I is constant, and, aside from a factor of 2π is exactly Ehrenfest's J. In this case I is not only an adiabatic invariant, it is rigorously constant. Since

1.44

if is independent of time, is independent of time as shown by eqn 1.36,

and thus is a constant of the motion. We can solve for I

1.45

1FK H I Isin 2t 2

⎡ ⎤∂ β= + = ω + γ⎢ ⎥∂ βω⎣ ⎦

&

0β =&

KK, 0⎧ ⎫ =⎨ ⎬ω⎩ ⎭Kω

KI(1 sin 2 )

=ω + α γ

2β

α ≡βω

&

Kω


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Transform to the new Hamiltonian K.

generating function:

eqn A.2

and use the inverse transformation eqns

so that

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2

F q tan 1K H It 2

⎛ ⎞∂ − γ −= + = ω + β⎜ ⎟∂ β⎝ ⎠

&

22

2 2

qK I tan I q tan2 2

⎡ ⎤β β= ω + γ = ω + γ⎢ ⎥β β ω⎣ ⎦

& &

1 ptanq

− βγ = − 2Iq 2I cos ,p sin= β γ = − γ

β

qp 2Isin cos Isin 2= − γ γ = − γ

2 221

jF q d qP tan (1 tan ) I

2 d 2∂

− = = γ = + γ =∂γ β γ β

1i

i

F qp tanq

∂= = − γ

∂ β

2

1q tanF

2γ

= −β


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so that the second term in A.2 becomes

therefore

p.s. solutions to dynamical problems almost always involve finding constants of the motion.Back to lecture p. 25

22

qpq tan Isin 22 2 2

⎛ ⎞β β − βγ = = γ⎜ ⎟β ω ω β βω⎝ ⎠

& & &

K I Isin 22

⎡ ⎤β= ω + γ⎢ ⎥βω⎣ ⎦

&


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and integrate the equation for γ

1.46

to find (see table of integrals)

1.47

1.48

d dt1 sin 2

γ= ω

+ α γ

1 02 2

tan2C tan1 1

− ⎛ ⎞γ + α= ⎜ ⎟

− α − α⎝ ⎠

21 2 1tan 1 tan C dt

2−

⎡ ⎤−α ⎡ ⎤γ =−α+ −α + ω⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦∫

KI

∂⎡ ⎤γ =⎢ ⎥∂⎣ ⎦&


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We see that I undergoes an oscillation whose amplitude is determined by α. ( If a beam with all values of γ and all values of I < Im undergoes such motion, we would term it a "bunch shape oscillation".) If α > 1, γ does not rotate around the origin, (because γ’ goes to zero) but is trapped in half the phase plane. Thus Symon's condition is just a limitation on the bunch shape oscillation amplitude. Further, if we choose the interval such that the phase changes by exactly π/2, then there is no change in I at the end of the process. Experimentally, one usually has control of the initial value of β, the final value of β, and the coefficient A (the rate of change) in 1.38. It is an easy matter experimentally to find the value of the parameters which yield minimal dilution.

End of Lecture


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Bohr-Sommerfeld theory of the atom (From X-Rays to Quarks, Emilio Segre)

experiments by Balmer deduces: - Rydberg constantfrom hydrogen atomic spectra

used plus constraints by on radii-> stationary states.

Bohr:1. “Dynamical equilibrium of the systems in the stationary states can be discussed by

the help of the ordinary mechanics, while the passing of the system between different stationary states cannot be treated on that basis.

2. That the latter process is followed by the emission of a homogeneous radiation, for which the relation between frequency and the amount of energy emitted is the one given by Planck’s theory (i.e., E1-E2=hν).”

Used correspondence principle to “match” at boundaries of transition from macroscopic to microscopic systems.

Results: allowed orbits

2 2

2

mv er r

=

2 2

n 2 2

n hr4 mZe

=π

2

n 2n

Ze 1E2r n

= − ∝

2 21 2

1 1n n

⎛ ⎞ν = ℜ −⎜ ⎟

⎝ ⎠

L n= h

ℜ


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derivations for Lecture 1

physics 598acc accelerators: theory and applicationsweb.hep.uiuc.edu/home/derrede/598acc/lecture 1...

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