Introduction to
Hamiltonian systems
Marlis Hochbruck
Heinrich-Heine Universitat Dusseldorf
Oberwolfach Seminar, November 2008
ExamplesMathematical biology: Lotka-Volterra modelFirst numerical methodsMathematical pendulumKepler problemOuter solar systemMolecular dynamics
First integralsEnergy, linear invariantsQuadratic and polynomial invariants
Reversible differential equationsSymmetric methods
Lotka-Volterra model I
◮ u(t) number of predators
◮ v(t) number of prey
u = u(v − 2)
v = v(1− u)
general autonomous system of odes
y = f (y)
◮ y point in phase space
◮ f (y) vector field (velocity in y)
◮ flow: ϕt : y0 7→ y(t) if y(0) = y0
Lotka-Volterra model II
1 2 3
1
2
3
4
5
1 2 3
1
2
3
4
5
1 2 3
1
2
3
4
5v
u
v
u
v
u
u number of predators, v number of prey
Invariant of Lotka-Volterra model
equationsu = u(v − 2), v = v(1− u)
divide by each other and separation of variables
0 =1− u
uu − v − 2
vv =
d
dtI (u, v)
with invariantI (u, v) = ln u − u + 2 ln v − v
◮ every solution lies on level curve of I
◮ level curves are closed thus all solutions are periodic
First numerical methods
autonomous problem y ′ = f (y)
◮ explicit Euler method:
yn+1 = yn + hf (yn)
◮ implicit Euler method:
yn+1 = yn + hf (yn+1)
◮ implicit midpoint rule
yn+1 = yn + hf
(yn + yn+1
2
)discrete or numerical flow: Φh : yn 7→ yn+1
Partitioned systems – symplectic Euler
partitioned system
u = f (u, v), v = g(u, v)
combine explicit and implicit Euler: symplectic Euler
un+1 = un + hf (un, vn+1)
vn+1 = vn + hg(un, vn+1)(SE1)
or
un+1 = un + hf (un+1, vn)
vn+1 = vn + hg(un+1, vn)(SE2)
SE1 becomes explicit if f (u, v) = f (u), g(u, v) = g(v)SE2 becomes explicit if f (u, v) = f (v), g(u, v) = g(u)
Lotka-Volterra model –experiment
2 4
2
4
6
2 4
2
4
6
2 4
2
4
6
v
u
v
u
v
u
explicit Euler implicit Euler symplectic Euler
y0
y82
y83
y0
y49
y50
y0y0
Hamiltonian problem
◮ Hamiltonian H(p, q) = H(p1, . . . , pd , q1, . . . , qd)(total energy)
◮ q1, . . . , qd positions
◮ p1, . . . , pd momenta
◮ Hamiltonian equations of motion
p = −Hq, Hq = ∇qH =
(∂H
∂q
)T
q = Hp
◮ energy conservation: H(p(t), q(t)) = const for all t
Mathematical pendulum
◮ mass m = 1, massless rod of length ℓ = 1,gravitational acceleration g = 1
◮ Hamiltonian
H(p, q) =1
2p2 − cos q
◮ equations of motion p = −Hq, q = Hp
p = − sin q, q = p
orq = − sin q
q
cos q ℓ
m
◮ vector field 2π-periodic in q =⇒ phase space cylinder R× S1
◮ flow ϕt(p, q) is an area preserving mapping
Area preservation Pendulum – numerical experiment
explicit Eulerh = 0.2
symplectic Eulerh = 0.3
Stormer-Verleth = 0.6
Kepler problem – two-body problem
◮ 1st body as center of coordinate system
◮ (p, q) coordinates of second body
Hamiltonian
H(p1, p2, q1, q2) =1
2(p2
1 + p22)− (q2
1 + q22)
−1/2
equations of motion:
qi = pi , pi = −Hqi = −qi (q21 + q2
2)−3/2
first integrals
◮ total energy H(p, q)
◮ angular momentum L(p1, p2, q1, q2) = q1p2 − q2p1
(Kepler’s second law)
Numerical example – Kepler problem
−2 −1 1
−1
−2 −1 1
1
−2 −1 1
−1
−2 −1 1
−1
1
400 000 stepsh = 0.0005
explicit Euler4 000 stepsh = 0.05symplectic Euler
4 000 stepsh = 0.05
implicit midpoint
4 000 stepsh = 0.05Stormer Verlet
Numerical example – Kepler problem II
50 100
.01
.02
50 100
.2
.4
conservation of energy
explicit Euler, h = 0.0001
symplectic Euler, h = 0.001
global error of solution
explicit Euler, h = 0.0001
symplectic Euler, h = 0.001
Qualitative long-time behavior – Kepler problem
method error in H error in L global error
explicit Euler O(th) O(th) O(t2h)
symplectic Euler O(h) 0 O(th)
implicit midpoint O(h2) 0 O(th2)
Stormer-Verlet O(h2) 0 O(th2)
Outer solar system
Hamiltonian
H(p, q) =1
2
5∑i=0
1
mipTi pi − g
5∑i=1
i−1∑j=0
mimj
‖qi − qj‖
◮ astronomical units (1 A.U. = 149 597 870 km)
◮ masses relative to mass of sun
◮ m0 = 1.00000597682 (account for inner planets)
◮ g = 2.95 . . . · 10−4 gravitational constant
◮ initial positions and initial velocity from Sept. 5, 1994, 0h00
Outer solar system – numerical example
explicit Euler, h = 10 implicit Euler, h = 10
symplectic Euler, h = 100 Stormer Verlet, h = 200
Molecular dynamics
Hamiltonian
H(p, q) =1
2
N∑i=1
1
mipTi pi +
N∑i=2
i−1∑j=1
Vij
(‖qi − qj‖
)
◮ Vij(r) potential function
◮ qi , pi positions and momenta of atoms
◮ mi atomic mass of ith atom
in molecular dynamics: Vij Lennard-Jones potential
Vij(r) = 4εij
((σij
r
)12 −(σij
r
)6)
3 4 5 6 7 8
−.2
.0
.2
Numerical experiment – frozen argon crystal
N = 7 argon atoms in a plane
1
2
3
4
5
6
7
temperature T =1
NkB
N∑i=1
mi‖qi‖2
Numerical experiment – argon crystal
−60
−30
0
30
60
−30
0
30−30
0
30
−60
−30
0
30
60
−30
0
30−30
0
30
explicit Euler, h = 0.5[fs]
symplectic Euler, h = 10[fs]
total energy
Verlet, h = 40[fs]
Verlet, h = 80[fs]
total energy
explicit Euler, h = 10[fs]
symplectic Euler, h = 10[fs]
temperature
Verlet, h = 10[fs]
Verlet, h = 20[fs]
temperature
First integrals
Definition. A non-constant function I (y) is called a first integral ofy = f (y) if
I ′(y)f (y) = 0 for all y .
synonyms: invariant, conserved quantity, constant of motion
Examples of first integrals
◮ total energy H(p, q) in Hamiltonian systems
◮ total linear and angular momentum of N-body systems
H(p, q) =1
2
N∑i=1
1
mipTi pi +
N∑i=2
i−1∑j=1
Vij(rij), rij = ‖qi − qj‖
equations of motion
qi =1
mipi , pi =
N∑j=1
νij(qi − qj), νij = −V ′ij(rij)/rij
◮ linear invariants I (y) = dT y , d constant, s.t. dT f (y) = 0
Quadratic and polynomial invariants
consider
Y = A(Y )Y , A(Y ) skew symmetric for all Y
where Y is a vector or a matrix
Theorem. The quadratic function I (Y ) = Y TY is invariant. Inparticular, orthogonality of Y0 is conserved.
Lemma. Let Y , A(Y ) ∈ Rn,n. If traceA(Y ) = 0 for all Y , thendetY is an invariant.
◮ det Y represents volume of parallelepiped generated bycolumns of Y
◮ volume convervation for traceA(Y ) = 0
Reversible differential equations
Definition. Let ρ be an invertible linear transformation in the phasespace of y = f (y). The differential equation and the vector fieldf (y) are called ρ-reversible if
ρf (y) = −f (ρy) for all y
u
v
−ρf (y)f (ρy)
ρ
y
ρy
f (y)
ρf (y)
u
v ϕt
ϕt
ρ ρ
y0
ρy0
y1
ρy1
Reversible vector fields – examples
◮ partitioned system
u = f (u, v), v = g(u, v)
where
f (u,−v) = −f (u, v), g(u,−v) = g(u, v)
is (ρ)-reversible for ρ(u, v) = (u,−v)
◮ second order differential equations
u = g(u) ⇐⇒ u = v , v = g(u)
are (ρ)-reversible
Do numerical methods produce a reversible numerical flow whenapplied to a reversible differential equation?
Symmetric methods
Definition. A numerical one-step method Φh is symmetric or timereversible if
Φh ◦ Φ−h = id.
y1 = Φh(y0) is symmetric if exchanging
y0 ↔ y1 and h ↔ −h
leaves the method unaltered
Examples: implicit midpoint rule, Stormer-Verlet method
Theorem. If a numerical method applied to a ρ-reversibledifferential equations satisfies
ρ ◦ Φh = Φ−h ◦ ρ
then Φh is ρ-reversible if and only if Φh is a symmetric method.