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Chapter 8
Hamiltonian dynamics
Conservative mechanical systems have equations of mo- tion that are symplectic and can be expressed in Hamilto- nian form. The generic properties within the class of sym- plectic vector fields are quite different from those within the class of all smooth vector fields: the system always has a first integral (“energy”) and a preserved volume, and equilibrium points can never be asymptotically stable in their energy level. — John Guckenheimer
Y
ou might think that the strangeness of contracting flows, flows such as the
Rössler flow of figure 2.6 is of concern only to chemists or biomedical
engineers or the weathermen; physicists do Hamiltonian dynamics, right?
Now, that’s full of chaos, too! While it is easier to visualize aperiodic dynamics
when a flow is contracting onto a lower-dimensional attracting set, there are plenty
of examples of chaotic flows that do preserve the full symplectic invariance of
Hamiltonian dynamics. The whole story started with Poincaré ’s restricted 3-
body problem, a realization that chaos rules also in general (non-Hamiltonian)
flows came much later.
Here we briefly review parts of classical dynamics that we will need later
on; symplectic invariance, canonical transformations, and stability of Hamiltonian
flows. If your eventual destination are applications such as chaos in quantum
and/or semiconductor systems, read this chapter. If you work in neuroscience
or fluid dynamics, skip this chapter, continue reading with the billiard dynamics
of chapter 9 which requires no incantations of symplectic pairs or loxodromic
quartets.
fast track: chapter 9, p. 155
8.1 Hamiltonian flows
“· · · to do this business right is a thing of far greater diffi- culty than I was aware of.” — Sir Isaac Newton, in a letter to Edmund Halley
(P. Cvitanovi´c and L.V. Vela-Arevalo)
An important class of flows are Hamiltonian flows, given by a Hamiltonian
appendix A
H(q, p) together with the Hamilton’s equations of motion
remark 2.
q ˙i =
∂H
∂pi
, p˙i = −
∂H
∂qi
with the d = 2 D phase-space coordinates x split into the configuration space
coordinates and the conjugate momenta of a Hamiltonian system with D degrees
of freedom (dof):
x = (q, p) , q = (q 1 , q 2 ,... , qD) , p = (p 1 , p 2 ,... , pD). (8.2)
The equations of motion (8.1) for a time-independent, D-dof Hamiltonian can be
written compactly as
x˙i = ωi jH, j(x) , H, j(x) =
∂x j
H(x) , (8.3)
where x = (q, p) ∈ M is a phase-space point, and the a derivative of (·) with
respect to x j is denoted by comma-index notation (·), j,
[
0 I
−I 0
]
is an antisymmetric [d×d] matrix, and I is the [D×D] unit matrix.
The energy, or the value of the time-independent Hamiltonian function at the
state space point x = (q, p) is constant along the trajectory x(t),
d
dt
H(q(t), p(t)) =
∂H
∂qi
q ˙i(t) +
∂H
∂pi
p ˙i(t)
∂H
∂qi
∂H
∂pi
∂H
∂pi
∂H
∂qi
so the trajectories lie on surfaces of constant energy, or level sets of the Hamilto-
nian {(q, p) : H(q, p) = E}. For 1-dof Hamiltonian systems this is basically the
whole story.
example 8. p. 149
Thus all 1-dof systems are integrable, in the sense that the entire phase plane
is stratified by curves of constant energy, either periodic, as is the case for the
if it preserves the symplectic bilinear form 〈 xˆ|x〉 = xˆ>ωx, where g>^ denotes the
transpose of g, and ω is a non-singular [2D × 2 D] antisymmetric matrix which
satisfies
remark 8.
ω>^ = −ω , ω^2 = − 1. (8.7)
While these are defining requirements for any symplectic bilinear form, ω is often
conventionally taken to be of form (8.4).
example 8. p. 149
If g is symplectic, so is its inverse g−^1 , and if g 1 and g 2 are symplectic, so
is their product g 2 g 1. Symplectic matrices form a Lie group called the symplec-
tic group Sp(d). Use of the symplectic group necessitates a few remarks about
Lie groups in general, a topic that we study in more depth in chapter 12. A Lie
group is a group whose elements g(φ) depend smoothly on a finite number N of
parameters φa. In calculations one has to write these matrices in a specific basis,
and for infinitesimal transformations they take form (repeated indices are summed
throughout this chapter, and the dot product refers to a sum over Lie algebra gen-
erators):
g(δφ) ' 1 + δφ · T , δφ ∈ RN^ , |δφ| 1 , (8.8)
where {T 1 , T 2 · · · , TN }, the generators of infinitesimal transformations, are a set
of N linearly independent [d×d] matrices which act linearly on the d-dimensional
phase space M. The infinitesimal statement of symplectic invariance follows by
substituting (8.8) into (8.6) and keeping the terms linear in δφ,
Ta>ω + ωTa = 0. (8.9)
This is the defining property for infinitesimal generators of symplectic transfor-
mations. Matrices that satisfy (8.9) are sometimes called Hamiltonian matrices.
A linear combination of Hamiltonian matrices is a Hamiltonian matrix, so Hamil-
tonian matrices form a linear vector space, the symplectic Lie algebra sp(d). By
the antisymmetry of ω,
(ωTa)>^ = ωTa. (8.10)
is a symmetric matrix. Its number of independent elements gives the dimen-
sion (the number of independent continuous parameters) of the symplectic group
Sp(d),
N = d(d + 1)/ 2 = D(2D + 1). (8.11)
The lowest-dimensional symplectic group Sp(2), of dimension N = 3, is isomor-
phic to SU(2) and SO(3). The first interesting case is Sp(3) whose dimension is
N = 10.
It is easily checked that the exponential of a Hamiltonian matrix
g = eφ·T^ (8.12)
is a symplectic matrix; Lie group elements are related to the Lie algebra elements
by exponentiation.
example ?? p. ??
8.3 Stability of Hamiltonian flows
Hamiltonian flows offer an illustration of the ways in which an invariance of equa-
tions of motion can affect the dynamics. In the case at hand, the symplectic in-
variance will reduce the number of independent Floquet multipliers by a factor of
2 or 4.
8.3.1 Canonical transformations
The evolution of Jt^ (4.5) is determined by the stability matrix A, (4.10):
d
dt
Jt(x) = A(x)Jt(x) , Ai j(x) = ωik H,k j(x) , (8.13)
where the symmetric matrix of second derivatives of the Hamiltonian, H,kn =
∂k∂nH, is called the Hessian matrix. From (8.13) and the symmetry of H,kn it
follows that for Hamiltonian flows (8.3)
A>ω + ωA = 0. (8.14)
This is the defining property (8.9) for infinitesimal generators of symplectic (or
canonical) transformations.
Consider now a smooth nonlinear coordinate change form yi = hi(x) (see
sect. 2.3 for a discussion), and define a ‘Kamiltonian’ function K(x) = H(h(x)).
Under which conditions does K generate a Hamiltonian flow? In what follows we
will use the notation ∂˜j = ∂/∂y j, si, j = ∂hi/∂x j. By employing the chain rule we
have that
K, j = H,˜l sl˜, j (8.15)
(Here, as elsewhere in this book, a repeated index implies summation.) By virtue
of (8.1), ˜∂lH = −ωlm y˙m, so that, again by employing the chain rule, we obtain
ωi j∂ jK = −ωi j s j,lωlm sm,n x˙n (8.16)
The right hand side simplifies to ˙xi (yielding Hamiltonian structure) only if
−ωi j sl, jωlm sm,n = δin (8.17)
or, in compact notation,
−ω(∂h)>ω(∂h) = 1 (8.18)
Figure 8.3: Stability of a symplectic map in R^4.
complex saddle saddle−center
degenerate saddle
(2) (2)
real saddle
generic center degenerate center
(2)
(2)
so the characteristic polynomial is reflexive, namely it satisfies
det (J − Λ 1 ) = det (J>^ − Λ 1 ) = det (−ωJ>ω − Λ 1 )
= det (J−^1 − Λ 1 ) = det (J−^1 ) det ( 1 − ΛJ)
= Λ^2 D^ det (J − Λ−^11 ). (8.21)
Hence if Λ is an eigenvalue of J, so are 1/Λ, Λ∗^ and 1/Λ∗. Real eigenvalues
always come paired as Λ, 1/Λ. The Liouville conservation of phase-space vol-
umes (8.30) is an immediate consequence of this pairing up of eigenvalues. The
complex eigenvalues come in pairs Λ, Λ∗, |Λ| = 1, or in loxodromic quartets Λ,
1 /Λ, Λ∗^ and 1/Λ∗. These possibilities are illustrated in figure 8.3.
example 8. p. 150
example 8. p. 150
example 8. p. 151
8.5 Poincaré invariants
Let C be a region in phase space and V(0) its volume. Denoting the flow of the
Hamiltonian system by f t(x), the volume of C after a time t is V(t) = f t(C), and
using (8.30) we derive the Liouville theorem:
V(t) =
f t(C)
dx =
C
∣∣det^
∂ f t(x′)
∂x
∣∣ dx
′
C
det (J)dx′^ =
C
dx′^ = V(0) , (8.22)
Hamiltonian flows preserve phase-space volumes.
The symplectic structure of Hamilton’s equations buys us much more than
the ‘incompressibility’, or the phase-space volume conservation. Consider the
symplectic product of two infinitesimal vectors
〈δx|δ xˆ〉 = δx>ωδ xˆ = δpiδ qˆi − δqiδ pˆi
∑^ D
i= 1
oriented area in the (qi, pi) plane
Time t later we have
〈δx′|δ xˆ′〉 = δx>^ J>ωJδ xˆ = δx>ωδ xˆ.
This has the following geometrical meaning. Imagine that there is a reference
phase-space point. Take two other points infinitesimally close, with the vectors δx
and δ xˆ describing their displacements relative to the reference point. Under the
dynamics, the three points are mapped to three new points which are still infinites-
imally close to one another. The meaning of the above expression is that the area
of the parallelepiped spanned by the three final points is the same as that spanned
by the initial points. The integral (Stokes theorem) version of this infinitesimal
area invariance states that for Hamiltonian flows the sum of D oriented areas Vi
bounded by D loops ΩVi, one per each (qi, pi) plane, is conserved:
V
d p ∧ dq =
ΩV
p · dq = invariant. (8.24)
One can show that also the 4, 6, · · · , 2 D phase-space volumes are preserved.
The phase space is 2D-dimensional, but as there are D coordinate combinations
conserved by the flow, morally a Hamiltonian flow is D-dimensional. Hence for
Hamiltonian flows the key notion of dimensionality is D, the number of the de-
grees of freedom (dof), rather than the phase-space dimensionality d = 2 D.
Dream student Henriette Roux: “Would it kill you to draw some pictures here?”
A: “Be my guest.”
in depth: appendix A8.1, p. 845
Résumé
Physicists do Lagrangians and Hamiltonians. Many know of no world other
than the perfect world of quantum mechanics and quantum field theory in which
the energy and much else is conserved. From the dynamical point of view, a
Hamiltonian flow is just a flow, but a flow with a symmetry: the stability matrix
Ai j = ωik H,k j(x) of a Hamiltonian flow ˙xi = ωi jH, j(x) satisfies A>ω + ωA = 0. Its
integral along the trajectory, the linearization of the flow J that we call the ‘Jaco-
bian matrix’, is symplectic, and a Hamiltonian flow is thus a canonical transforma-
tion in the sense that the Hamiltonian time evolution x′^ = f t(x) is a transformation
whose linearization (Jacobian matrix) J = ∂x′/∂x preserves the symplectic form,
maximal possible rank. The odd symplectic groups Sp(2D + 1) are not semisimple. If you care about group theory for its own sake (the dynamical systems symmetry reduction techniques of chapter 12 are still too primitive to be applicable to Quantum Field Theory), chapter 14 of ref. [4] is fun, too.
Referring to the Sp(d) Lie algebra elements as ‘Hamiltonian matrices’ as one some- times does [5, 28] conflicts with what is meant by a ‘Hamiltonian matrix’ in quantum mechanics: the quantum Hamiltonian sandwiched between vectors taken from any com- plete set of quantum states. We are not sure where this name comes from; Dragt cites refs. [8, 10], and chapter 17 of his own book in progress [6]. Fulton and Harris [8] use it. Certainly Van Loan [22] uses in 1981, and Taussky in 1972. Might go all the way back to Sylvester?
Dream student Henriette Roux wants to know: “Dynamics equals a Hamiltonian plus a bracket. Why don’t you just say it?” A: “It is true that in the tunnel vision of atomic mechanicians the world is Hamiltonian. But it is much more wondrous than that. This chapter starts with Newton 1687: force equals acceleration, and we always replace a higher order time derivative with a set of first order equations. If there are constraints, or fully relativistic Quantum Field Theory is your thing, the tool of choice is to recast New- ton equations as a Lagrangian 1788 variational principle. If you still live in material but non-relativistic world and have not gotten beyond Heisenberg 1925, you will find Hamil- ton’s 1827 principal function handy. The question is not whether the world is Hamiltonian
- it is not - but why is it so often profitably formulated this way. For Maupertuis 1744 vari- ational principle was a proof of God’s existence; for Lagrange who made it mathematics, it was just a trick. Our sect. 37.1.1 “Semiclassical evolution” is an attempt to get inside 17 year old Hamilton’s head, but it is quite certain that he did not get to it the way we think about it today. He got to the ‘Hamiltonian’ by studying optics, where the symplectic struc- ture emerges as the leading WKB approximation to wave optics; higher order corrections destroy it again. In dynamical systems theory, the densities of trajectories are transported by Liouville evolution operators, as explained here in sect. 19.6. Evolution in time is a one-parameter Lie group, and Lie groups act on functions infinitesimally by derivatives. If the evolution preserves additional symmetries, these derivatives have to respect them, and so ‘brackets’ emerge as a statement of symplectic invariance of the flow. Dynamics with a symplectic structure are just a special case of how dynamics moves densities of tra- jectories around. Newton is deep, Poisson brackets are technology and thus they appear naturally only by the time we get to chapter 19. Any narrative is of necessity linear, and putting Poisson ahead of Newton [27] would be a disservice to you, the student. But if you insist: Dragt and Habib [5, 7] offer a concise discussion of symplectic Lie operators and their relation to Poisson brackets. ”
Remark 8.2. Symplectic. The term symplectic –Greek for twining or plaiting
together– was introduced into mathematics by Hermann Weyl. ‘Canonical’ lineage is
church-doctrinal: Greek ‘kanon’, referring to a reed used for measurement, came to mean
in Latin a rule or a standard.
Remark 8.3. The sign convention of ω. The overall sign of ω, the symplectic invariant
in (8.3), is set by the convention that the Hamilton’s principal function (for energy con-
serving flows) is given by R(q, q′, t) =
∫ (^) q′ q pidqi^ −^ Et. With this sign convention the action along a classical path is minimal, and the kinetic energy of a free particle is positive. Any finite-dimensional symplectic vector space has a Darboux basis such that ω takes form (8.6). Dragt [5] convention for phase-space variables is as in (8.2). He calls the dynam- ical trajectory x 0 → x(x 0 , t) the ‘transfer map’, something that we will avoid here, as it conflicts with the well established use of ‘transfer matrices’ in statistical mechanics.
Remark 8.4. Loxodromic quartets. For symplectic flows, real eigenvalues always
come paired as Λ, 1/Λ, and complex eigenvalues come either in Λ, Λ∗^ pairs, |Λ| = 1, or
Λ, 1/Λ, Λ∗, 1/Λ∗^ loxodromic quartets. As most maps studied in introductory nonlinear
dynamics are 2d, you have perhaps never seen a loxodromic quartet. How likely are we to run into such things in higher dimensions? According to a very extensive study of
periodic orbits of a driven billiard with a four dimensional phase space, carried in ref. [16], the three kinds of eigenvalues occur with about the same likelihood.
Remark 8.5. Standard map. Standard maps model free rotors under the influence
of short periodic pulses, as can be physically implemented, for instance, by pulsed optical
lattices in cold atoms physics. On the theoretical side, standard maps illustrate a number of important features: small k values provide an example of KAM perturbative regime
(see ref. [14]), while larger k’s illustrate deterministic chaotic transport [3, 18], and the transition to global chaos presents remarkable universality features [12, 13, 25]. The
quantum counterpart of this model has been widely investigated, as the first example
where phenomena like quantum dynamical localization have been observed [1]. Stability residue was introduced by Greene [12]. For some hands-on experience of the standard
map, download Meiss simulation code [19].
Remark 8.6. Diagnosing chaos. In sect. 1.3.1 we have stated that a deterministic
system exhibits ‘chaos’ if its orbits are locally unstable (positive Lyapunov exponent) and globally mixing (positive entropy). In sect. 6.2 we shall define Lyapunov exponents and
discuss their evaluation, but already at this point it would be handy to have a few quick nu- merical methods to diagnose chaotic dynamics. Laskar’s frequency analysis method [15]
is useful for extracting quasi-periodic and weakly chaotic regions of state space in Hamil- tonian dynamics with many degrees of freedom. For pointers to other numerical methods,
see ref. [26].
References
[1] G. Casati and B. V. Chirikov, Quantum Chaos: Between Order and Disor-
der (Cambridge Univ. Press, Cambridge, 1995).
[2] T. M. Cherry, “Some examples of trajectories defined by differential equa-
tions of a generalized dynamical type”, Trans. Camb. Phil. Soc. 23 , 165–
[3] B. V. Chirikov, “A universal instability of many-dimensional oscillator sys-
tem”, Phys. Rep. 52 , 263–379 (1979).
[4] P. Cvitanovi´c, Group Theory - Birdtracks, Lie’s, and Exceptional Groups
(Princeton Univ. Press, Princeton, NJ, 2008).
[5] A. J. Dragt, “The symplectic group and classical mechanics”, Ann. New
York Acad. Sci. 1045 , 291–307 (2005).
[6] A. J. Dragt, Lie methods for nonlinear dynamics with applications to accel-
erator physics, 2011.
[7] A. J. Dragt and S. Habib, How Wigner functions transform under symplec-
tic maps, 1998.
[8] W. Fulton and J. Harris, Representation Theory (Springer, New York, 1991).
Example 8.1. Unforced undamped Duffing oscillator. When the damping term is removed from the Duffing oscillator (2.22), the system can be written in Hamiltonian form,
H(q, p) =
p^2 2
q^2 2
q^4 4
This is a 1-dof Hamiltonian system, with a 2-dimensional state space, the plane (q, p). The Hamilton’s equations (8.1) are
q˙ = p , p˙ = q − q^3. (8.26)
For 1-dof systems, the ‘surfaces’ of constant energy (8.5) are curves that stratify the phase
plane (q, p), and the dynamics is very simple: the curves of constant energy are the tra- jectories, as shown in figure 8.1. click to return: p. 139
Example 8.2. Collinear helium. In the quantum chaos part of ChaosBook.org we shall apply the periodic orbit theory to the quantization of helium. In particular, we will study collinear helium, a doubly charged nucleus with two electrons arranged on a line, an electron on each side of the nucleus. The Hamiltonian for this system is
H =
p^21 +
p^22 −
r 1
r 2
r 1 + r 2
Collinear helium has 2 dof, and thus a 4-dimensional phase space M, which energy con-
servation stratified by 3-dimensional constant energy hypersurfaces. In order to visualize it, we often project the dynamics onto the 2-dimensional configuration plane, the (r 1 , r 2 ),
ri ≥ 0 quadrant, figure 8.2. It looks messy, and, indeed, it will turn out to be no less chaotic than a pinball bouncing between three disks. As always, a Poincaré section will
be more informative than this rather arbitrary projection of the flow. The difference is that
in such projection we see the flow from an arbitrary perspective, with trajectories criss- crossing. In a Poincaré section the flow is decomposed into intrinsic coordinates, a pair
along the marginal stability time and energy directions, and the rest transverse, revealing the phase-space structure of the flow. click to return: p. 140
Example 8.3. Symplectic form for D = 2. For two degrees of freedom the phase space is 4-dimensional, x = (q 1 , q 2 , p 1 , p 2 ) , and the symplectic 2-form is
ω =
The symplectic bilinear form 〈x(1)|x(2)〉 is the sum over the areas of the parallelepipeds spanned pairwise by components of the two vectors,
〈x(1)|x(2)〉 = (x(1))>ω x(2)^ = (q(1) 1 p(2) 1 − q(2) 1 p(1) 1 ) + (q(1) 2 p(2) 2 − q(2) 2 p(1) 2 ). (8.29)
It is this sum over oriented areas (not the Euclidean distance between the two vectors, |x(2)^ − x(1)|) that is preserved by the symplectic transformations. click to return: p. 141
Example 8.4. Hamiltonian flows are canonical. For Hamiltonian flows it follows from (8.14) that (^) dtd
J>ωJ
= 0, and since at the initial time J^0 (x 0 ) = 1 , Jacobian matrix is a symplectic transformation (8.6). This equality is valid for all times, so a Hamiltonian flow f t(x) is a canonical transformation, with the linearization ∂x f t(x) a symplectic trans- formation (8.6): For notational brevity here we have suppressed the dependence on time
Figure 8.4: Stability exponents of a Hamiltonian equi- librium point, 2-dof.
complex saddle saddle-center
degenerate saddle
(2) (2)
real saddle
generic center degenerate center
(2) (2)
and the initial point, J = Jt(x 0 ). By elementary properties of determinants it follows from (8.6) that Hamiltonian flows are phase-space volume preserving, |det J| = 1. The initial condition (4.10) for J is J^0 = 1 , so one always has
det J = + 1. (8.30)
click to return: p. 143
Example 8.5. Hamiltonian Hénon map, reversibility. By (4.45) the Hénon map (3.18) for b = −1 value is the simplest 2-dimensional orientation preserving area- preserving map, often studied to better understand topology and symmetries of Poincaré sections of 2 dof Hamiltonian flows. We find it convenient to multiply (3.19) by a and absorb the a factor into x in order to bring the Hénon map for the b = −1 parameter value into the form
xi+ 1 + xi− 1 = a − x^2 i , i = 1 , ..., np , (8.31)
The 2-dimensional Hénon map for b = −1 parameter value
xn+ 1 = a − x^2 n − yn yn+ 1 = xn. (8.32)
is Hamiltonian (symplectic) in the sense that it preserves area in the [x, y] plane.
For definitiveness, in numerical calculations in examples to follow we shall fix (arbi- trarily) the stretching parameter value to a = 6, a value large enough to guarantee that all
roots of 0 = f n(x) − x (periodic points) are real. exercise 9. click to return: p. 144 Example 8.6. 2-dimensional symplectic maps. In the 2-dimensional case the eigen- values (5.5) depend only on tr Mt
Λ 1 , 2 =
tr Mt^ ±
(tr Mt^ − 2)(tr Mt^ + 2)
Greene’s residue criterion states that the orbit is (i) elliptic if the stability residue |tr Mt| − 2 ≤ 0, with complex eigenvalues Λ 1 = eiθt, Λ 2 = Λ∗ 1 = e−iθt. If |tr Mt| − 2 > 0, λ is real, and the trajectory is either
(ii) hyperbolic Λ 1 = eλt^ , Λ 2 = e−λt^ , or (8.34) (iii) inverse hyperbolic Λ 1 = −eλt^ , Λ 2 = −e−λt^. (8.35)
click to return: p. 144
EXERCISES 153
and
G′(x) = −g(x). (8.40)
Important features of this map, including transition to global chaos (destruction of the last invariant torus), may be tackled by detailed investigation of the stability of periodic orbits. A family of periodic orbits of period Q already present in the k = 0 rotation maps can be labeled by its winding number P/Q The Greene residue describes the stability of a P/Q-cycle:
RP/Q =
2 − tr MP/Q
If RP/Q ∈ (0, 1) the orbit is elliptic, for RP/Q > 1 the orbit is hyperbolic orbits, and for RP/Q < 0 inverse hyperbolic.
For k = 0 all points on the y 0 = P/Q line are periodic with period Q, winding number P/Q and marginal stability RP/Q = 0. As soon as k > 0, only a 2Q of such orbits survive, according to Poincaré -Birkhoff theorem: half of them elliptic, and half hyperbolic. If
we further vary k in such a way that the residue of the elliptic Q-cycle goes through 1, a bifurcation takes place, and two or more periodic orbits of higher period are generated. click to return: p. 144
Exercises
8.1. Complex nonlinear Schrödinger equation. Con- sider the complex nonlinear Schrödinger equation in one spatial dimension [17]:
i
∂φ ∂t
∂^2 φ ∂x^2
(a) Show that the function ψ : R → C defining the traveling wave solution φ(x, t) = ψ(x−ct) for c > 0 satisfies a second-order complex differential equa- tion equivalent to a Hamiltonian system in R^4 rel- ative to the noncanonical symplectic form whose matrix is given by
wc =
− 1 0 0 −c 0 − 1 c 0
(b) Analyze the equilibria of the resulting Ha- miltonian system in R^4 and determine their linear stability properties. (c) Let ψ(s) = eics/^2 a(s) for a real function a(s) and determine a second order equation for a(s). Show
that the resulting equation is Hamiltonian and has heteroclinic orbits for β < 0. Find them. (d) Find ‘soliton’ solutions for the complex nonlinear Schrödinger equation.
(Luz V. Vela-Arevalo)
8.2. Symplectic vs. Hamiltonian matrices. In the language of group theory, symplectic matrices form the symplectic Lie group Sp(d), while the Hamiltonian ma- trices form the symplectic Lie algebra sp(d), or the al- gebra of generators of infinitesimal symplectic transfor- mations. This exercise illustrates the relation between the two:
(a) Show that if a constant matrix A satisfy the Hamil- tonian matrix condition (8.9), then J(t) = exp(tA) , t ∈ R, satisfies the symplectic condition (8.6), i.e., J(t) is a symplectic matrix. (b) Show that if matrices Ta satisfy the Hamiltonian matrix condition (8.9), then g(φ) = exp(φ · T) , φ ∈ RN^ , satisfies the symplectic condition (8.6), i.e., g(φ) is a symplectic matrix.
exerNewton - 13jun2008 ChaosBook.org edition16.0, Jan 13 2018
EXERCISES 154
(A few hints: (i) expand exp(A) , A = φ · T , as a power series in A. Or, (ii) use the linearized evolution equation (8.13). ) 8.3. When is a linear transformation canonical?
(a) Let A be a [n × n] invertible matrix. Show that the map φ : R^2 n^ → R^2 n^ given by (q, p) 7 → (Aq, (A−^1 )>p) is a canonical transformation. (b) If R is a rotation in R^3 , show that the map (q, p) 7 → (R q, R p) is a canonical transformation.
(Luz V. Vela-Arevalo) 8.4. Determinants of symplectic matrices. Show that the determinant of a symplectic matrix is +1, by going through the following steps:
(a) use (8.21) to prove that for eigenvalue pairs each member has the same multiplicity (the same holds for quartet members), (b) prove that the joint multiplicity of λ = ±1 is even, (c) show that the multiplicities of λ = 1 and λ = − 1 cannot be both odd. Hint: write
P(λ) = (λ − 1)^2 m+^1 (λ + 1)^2 l+^1 Q(λ)
and show that Q(1) = 0.
8.5. Cherry’s example. What follows refs. [2, 20] is mostly a reading exercise, about a Hamiltonian system that is linearly stable but nonlinearly unstable. Consider the Hamiltonian system on R^4 given by
H =
(q^21 + p^21 ) − (q^22 + p^22 ) +
p 2 (p^21 − q^21 ) − q 1 q 2 p 1.
(a) Show that this system has an equilibrium at the origin, which is linearly stable. (The linearized system consists of two uncoupled oscillators with frequencies in ratios 2:1). (b) Convince yourself that the following is a family of solutions parameterize by a constant τ:
q 1 = −
cos(t − τ) t − τ
, q 2 = cos 2(t − τ) t − τ
p 1 =
sin(t − τ) t − τ
, p 2 =
sin 2(t − τ) t − τ
These solutions clearly blow up in a finite time; however they start at t = 0 at a distance
3 /τ from the origin, so by choosing τ large, we can find solutions starting arbitrarily close to the ori- gin, yet going to infinity in a finite time, so the origin is nonlinearly unstable.
(Luz V. Vela-Arevalo)
exerNewton - 13jun2008 ChaosBook.org edition16.0, Jan 13 2018
