Introduction to Hamiltonian systems Marlis Hochbruck Heinrich-Heine Universit¨ at D¨ usseldorf Oberwolfach Seminar, November 2008 Examples Mathematical biology: Lotka-Volterra model First numerical methods Mathematical pendulum Kepler problem Outer solar system Molecular dynamics First integrals Energy, linear invariants Quadratic and polynomial invariants Reversible differential equations Symmetric 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: ϕ t : y 0 → y (t ) if y (0) = y 0 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 5 v u v u v u u number of predators, v number of prey Invariant of Lotka-Volterra model equations ˙ u = u(v − 2), ˙ v = v (1 − u) divide by each other and separation of variables 0= 1 − u u ˙ u − v − 2 v ˙ v = d dt I (u, v ) with invariant I (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: y n+1 = y n + hf (y n ) ◮ implicit Euler method: y n+1 = y n + hf (y n+1 ) ◮ implicit midpoint rule y n+1 = y n + hf y n + y n+1 2 discrete or numerical flow: Φ h : y n → y n+1 Partitioned systems – symplectic Euler partitioned system ˙ u = f (u, v ), ˙ v = g (u, v ) combine explicit and implicit Euler: symplectic Euler u n+1 = u n + hf (u n , v n+1 ) v n+1 = v n + hg (u n , v n+1 ) (SE1) or u n+1 = u n + hf (u n+1 , v n ) v n+1 = v n + hg (u n+1 , v n ) (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 y0 y0
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Introduction to Hamiltonian systems - KIT · symplectic Euler, h = 100 StoÄrmer Verlet, h = 200 Molecular dynamics Hamiltonian H (p;q) = 1 2 XN i=1 1 m i pT ip + XN i=2 Xi¡ 1 j=1
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◮ 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