1. 1. Review of Hamiltonian Dynamics 1.1. Principle of Least Action

2. 1. Review of Hamiltonian dynamics 1.1. Principle of “least action”
Historically, it was Newton’s law of dynamics (dating 1687) that opened the door for the analytic treatment of particle dynamics
Later Leibnitz (1707) and others (Maupertuis, Euler, Jacobi, Lagrange, Hamilton) formulated the more fundamental
principle of least action
from which Newton’s law can be derived.
It corresponds to Fermat’s principle of least time — from which Snell’s law of refraction of light is obtained.

3. 1. Review of Hamiltonian dynamics 1.1. Principle of “least action”
Lifesaver problem: how to find the route that takes the minimum time?

4. 1. Review of Hamiltonian dynamics 1.1. Principle of “least action”
Given: dynamical system of n degrees of freedom with the
n-dimensional vector q and covector p of generalized coordinates.
A path g : → 2n in the 2n dimensional phase space that is para-meterized by the time t as the independent variable is defined by
With H the system’s Hamiltonian, the principle of least action is formulated by means of the functional F(g) as the line integral
The task is to find gext that minimizes F(g), i.e. for which

5. 1. Review of Hamiltonian Dynamics 1.2. Canonical Equations

6. 1. Review of Hamiltonian dynamics 1.2. Canonical equations
In text form, the principle reads:
Among all thinkable paths g, a dynamical system “chooses” exactly that one gext, where F(gext) takes on a minimum.
In the picture of the rescuer, the system always takes the optimum path.
Max Planck’s observation: The principle applies for all reversible phenomena of physics.
Result of calculus of variations: we have dF(gext) = 0 exactly if the phase-space path satisfies the “canonical equations” (exercise!)
These equations are associated with a phase-space symmetry, which gives rise to corresponding conserved quantities (e.g. “Liouville”).

7. 1. Review of Hamiltonian dynamics 1.2. Canonical equations
A system is said to be conservative if there exists a scalar function V(q,t) such that the force F(q,t) acting on a particle is given by
For a conservative system, the Hamiltonian H(q,p,t) is given by the sum of the system’s kinetic energy T and its potential energy V. For a system of particles of the same mass m, we then have
The Hamiltonian thus represents
the system’s instantaneous total
energy – which is not a conserved
quantity if H is explicitly time dep.
The canonical equations derived from this Hamiltonian are:
 Newton’s equations of motion are reproduced with this Hamiltonian.

8. 1. Review of Hamiltonian Dynamics 1.3. Reversibility

9. 1. Review of Hamiltonian dynamics 1.3. Reversibility
For a given Hamiltonian, the equations of for the phase space trajectory (q(t),p(t)) are given by the canonical equations
We apply the time reversal transformation
 The canonical equations are invariant under this transformation:
 Hamiltonian dynamics describe reversible systems, no information is lost in the course of its time evolution. The system has an infinite memory: all previous states can be recovered.

10. 1. Review of Hamiltonian dynamics 1.3. Reversibility
Reversible system: the initial state is recovered after time reversal

11. 1. Review of Hamiltonian dynamics 1.3. Reversibility
Computer simulation of a reversible system: the initial state is not recovered due to information loss (here: because of finite accuracy)

12. Review of Hamiltonian dynamics 1.4. Canonical transformations

13. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Given: Hamiltonian H : 2n+1 →  of a system of n degrees of freedom with the n-dimensional vector q and covector p of generalized coordinates. The principle of least action is formulated by means of the functional F(g) along a phase-space path g
The specific dependence of the
value of H on each qi, pi and of
the time t provides the complete
information on the given system.
as the requirement to find gext that minimizes F(g), i.e. for which
Commonly, the phase-space path g : → 2n is parameterized by the time t as the independent variable

14. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
The line integral along g is thus converted into the ordinary integral
Result of calculus of variations: we have dF(gext) = 0 exactly if the phase-space path satisfies the “canonical equations”
Remark: if H depends explicitly on time t, the parameterization along the time is not the most general one. A more general parameter-ization of the line integral along g would be:
We will discuss this topic
later in the context of a
“generalized canonical
transformation theory”

15. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Clearly, we are not tied to a particular coordinate system if we want to treat a given dynamical system.
 We are always allowed to switch to another coordinate description
The Qi, Pi can represent either the
same system in other coordinates
or another system which is then
correlated to the given system.
In order for the transformed description to be equivalent to the original one, the principle of least action must be maintained
We observe that the time t is the common independent variable of both the original and the transformed system. The time scales of both systems coincide  special case!

16. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
The requirement that the integrals must agree means that the integrands may differ by the total time derivative of an arbitrary function F1, which is referred to as the generating function:
Reason: the term dF1/dt can be immediately integrated over t. The variation dF1 at the boundaries t1 and t2 vanishes by assumption
Adding dF1/dt to the integrand does not modify the variation integral.
 In principle, F1 can be a function of arbitrary arguments. With regard to the integrand equation, only F1 = F1(q,Q,t) makes sense.

17. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
With this particular argument list of F1, its total time derivative is
We can now compare the coefficients of dF1/dt with those of the integrand equation to obtain the following transformation rules
as the variation integral is maintained by this transformation, the canonical equation (which follow thereof) are equally maintained
This feature distinguishes canonical transformations from arbitrary transformations which do not conserve the form of the can. equations.

18. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
It is often advantageous to express the “generating function” of a canonical transformation in terms of the original space coordinates qi and the transformed momentum coordinates Pi. The transition from F1(q,Q,t) to a function F2(q,P,t) is accomplished by a Legendre transformation
 We can then derive an equivalent set of transformations rules. Obviously, the q- and the t-dependence of F1 and F2 coincide
 The corresponding transformation rules are the same. The new rule follows from the Q- dependence of F1 and the P-dependence of F2

19. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
In contrast to F2= F2(q,P,t), the generating function F1= F1(Q,q,t) does not depend on the Pi . The derivative of F2 with respect to Pi is thus simply given by
Summarizing, we obtain the complete set of transformation rules for F2
Moreover, two more representations of generating functions exist, namely F3= F3(Q,p,t) and F4= F4(P,p,t) – which lead to equivalent sets of transformation rules.

20. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Proceeding as above, by means of the Legendre transformation
we obtain the complete set of transformation rules for F3 as
Finally, the Legendre transformation
yields the transformation rules

21. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
From the four generating functions, the following four symmetry relations emerge calculating their second derivatives
The existence of these symmetry relations distinguishes a canonical transformation from a general coordinate transformation. For the latter, the symmetry relations do not apply.
We will make use of the symmetry relations to prove Liouville’s theorem.

22. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Direct solution approach versus canonical transformation approach: both are equivalent but often the direct way is not the easiest one!

23. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Simple examples:
For a harmonic oscillator, we can switch to action-angle variables. The solution of the transformed canonical equations becomes trivial.
A damped harmonic oscillator can be mapped into an undamped harmonic oscillator. We then only have to solve the equations of motion for the undamped case and obtain the solution for the damped oscillator by canonical (back) transformation.
A time-dependent damped harmonic oscillator can be canonically mapped into a conventional (time-independent) harmonic oscillator. This is an easy way to obtain the general solution for the time-dependent damped harmonic oscillator – which was solved for the first time around 1960 ! (Courant and Snyder, Theory of the alternating gradient synchrotron).

24. 1. Review of Hamiltonian dynamics 1.4. Canonical transformations
Exercise: The Hamiltonian of the damped harmonic oscillator is:
Verify that the canonical equations yield the equation of motion
and solve the equation of motion (q(0),p(0)) → (q(t),p(t)) by means of the canonical transformation generated by
The steps are: 1. derive the transformation rules and the Hamiltonian of the transformed system H′(Q,P,t), 2. find (Q(0),P(0)) → (Q(t),P(t)), 3. apply (q(0),p(0)) → (Q(0),P(0)) and (Q(t),P(t)) → (q(t),p(t)).

25. Review of Hamiltonian dynamics 1.5. Liouville’s theorem

26. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
Under a general coordinate transformation (i,j = 1,…,n)
the change of the volume form dV is determined by the determinant det J of the Jacobi matrix J
with

27. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
Given a Hamiltonian system H(q,p,t) of n degrees of freedom. The volume form induced by all variables dV =dq1…dqndp1…dpn is invariant under canonical transformations (CTs), i.e.
The proof of Liouville’s theorem thus means to show that det J = 1 if the transformation rules can be derived from a generating function.
We are going to prove this theorem by making use of the symmetry relations we derived from the four types of generating functions.

28. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
We first rewrite the Jacobi matrix J as 2×2 block matrix
with A,B,C, and D denoting the n×n matrices
The determinant of such a block matrix J is given by
The matrix S is referred to as the Schur complement of matrix A.

29. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
We now assume that the inverse coordinate transformation exists
hence
with

30. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
and rewrite the Jacobi matrix J-1 as the 2×2 block matrix
with E,F,G, and H denoting the n×n matrices
The determinant of the block matrix J-1 is given by
The matrix S′ is referred to as the Schur complement of matrix H.

31. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
For canonical coordinate transformations, the matrices E,F,G, and H are correlated with the matrices A,B,C, and D according to the symmetry relations that were derived in Sec. 1.4
We thus have
As det J is equal to its reciprocal, we conclude that det J = 1.

32. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
An infinitesimal canonical transformation that is generated by
pushes the Hamiltonian system H(q, p, t) one step dt forward in time. To first order, the transformation rules are (exercise):
We can repeat this transformation an arbitrary number of times.
 The mapping over finite time steps is also canonical.
 The system’s time evolution constitutes a canonical transformation.

33. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
Special case of Liouville’s theorem:
Given a Hamiltonian system H(q,p,t) of n degrees of freedom. The volume form induced by all variables dV =dq1…dqndp1…dpn is invariant under the system’s time evolution, i.e.
Interpretation: As the system moves forward in time, the system point (q1,…,qn,p1,…,pn) changes its location in the 2n-dimensional phase space. An infinitesimal volume around this point remains invariant.
 The local point density around a system point in the 2n-dimensional phase space remains invariant.

34. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
As an example, we consider the mathematical pendulum. Its Hamiltonian and the resulting equation of motion are
The system has one degree of freedom  the phase space has the dimension 2 and can hence be visualized.
We plot the time evolution of two sets of phase-space points, each of them representing a particular initial condition of the pendulum.
In the first case, the initial distribution of point is uniform, whereas in the second sequence the distribution of initial conditions is Gaussian.

35. 1. Review of Hamiltonian dynamics 1.5. Liouville’s theorem
Time evolution of a uniform (top) and a Gaussian (bottom) phase-space distribution of initial conditions for the mathematical pendulum.

36. Review of Hamiltonian dynamics 1.6. Discussion of Liouville’s Theorem

37. General case of charged particle beam optics (point particles!):
1. Review of Hamiltonian dynamics 1.6. Discussion of Liouville’s theorem
General case of charged particle beam optics (point particles!):
System of N interacting particles in the 3-d real space
 the system has 3N degrees of freedom
 the phase space has the 6N dimensions
 the entire system is represented by one point in this space
An ensemble of such systems forms a distribution of points in this 6N-dimensional phase space.
Liouville’s theorem then states that the local density of points in this 6N-dimensional phase space is conserved in time.
In this form, the theorem is not very helpful for our purposes!

38. Each of the N particles resides in its own 6-dim. phase space
1. Review of Hamiltonian dynamics 1.6. Discussion of Liouville’s theorem
What changes if the particle interaction forces are so weak that they can be neglected (“zero space-charge limit”) ?
Each of the N particles resides in its own 6-dim. phase space
 the 6N-dim. phase space factorizes into N copies of a 6-dim. phase space
 Liouville’s theorem holds separately for each particle in the 6-dim. phase space
 the set of N particles of our charged particle beam forms a distribution of points in this 6-dim. phase space
 Liouville’s theorem then states that the local point density of points in this 6-dimensional phase space is conserved in time.
 Note: this holds only if the particles do not interact!

39. What happens if the particle interaction forces are not weak ?
1. Review of Hamiltonian dynamics 1.6. Discussion of Liouville’s theorem
What happens if the particle interaction forces are not weak ?
 Then, the local point density in the 6-dim. phase space is not conserved
 In other words, Liouville’s theorem then cannot be applied to the 6-dim. phase space. We must either go back to the 6N-dim. phase space (no option),
 or, we must approximate the particle-particle interaction forces by a smooth field that is equivalent to an external field.
 Liouville’s theorem then states that the local point density of points in this 6-dimensional phase space is conserved in time.
 Note: all binary particle-particle interactions are neglected, the system of N charged particles is treated as if it were a continuum.

40. 1. Review of Hamiltonian dynamics 1. 6
1. Review of Hamiltonian dynamics 1.6. Discussion of Liouville’s theorem
In the continuum approximation, Liouville’s theorem holds in the 6-dim. phase space
 We have thus eliminated the characteristic feature of charged particle beams being made up of individual particles.
Yet, in reality, we encounter the effect of “intra-beam scattering”, which limits the life-time of any beam in a storage ring.
This effect is due to individual particle-particle interactions and shows that Liouville’s theorem holds only approximately in the 6-dim. phase space.
In the 2-dim. subspaces (x,x′), (y,y′), and (Δz, Δz′), Liouville’s theorem never holds strictly, but only in approximately if the motion in the respective plane is not coupled to the motion in the other planes.