Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton s laws of motion. Solving equations in Maple. Harmonic oscillator as a dynamical system. Dynamical systems and critical points. Chaotic systems.
Fundamental theories Classical mechanics Classical physics Thermodynamics and statistical physics Theory of electromagnetism Special theory of relativity Quantum mechanics General theory of relativity Quantum field theory??? (string theory, loop quantum gravity)
Revolutions of 20th century relativity of space and time (STR) time depends on observer quantization of energy (QM) stability of atoms gravitation is curvature of the space-time (GR) black holes, expansion of the Universe and Big Bang discovery of elementary particles (QFT=STR+QM) creation and annihilation of particles, antiparticles radiation of black holes quantum field theory on curved space-time acceleration of the expansion of the Universe unsolved problem
Yet another revolution: classical deterministic chaos Classical mechanics is wrong but useful Classical systems are deterministic, behaviour of the system is completely determined by initial conditions Some deterministic systems are extremely complicated fluids, Solar system difficulties 1. it is impossible to solve corresponding equations 2. it is impossible to measure initial conditions example Gas - 10 20 particles in cm 3
What chaos is not example: crystal of sodium chloride Na + Cl complicated system, complicated oscillatory motion of atoms impossible to solve equations (approx. methods: condensed matter physics and quantum mechanics) very stable system, almost insensitive to external conditions structure of lattice is determined only by ion-bound between Na + and Cl
Laminar flow of fluid Complicated system: big number of fluid particles Only for small velocities Unstable regime
Turbulent flow of fluid For higher velocities the motion becomes chaotic Unpredictable creation of eddies and vortices
Intuitive characteristics of chaos Unpredictable behaviour High sensitivity to initial conditions Unsolvable equations of motion Examples tubulence, weather convection of heat in the atmosphere damped, driven pendulum traffic economics Why study of chaos belongs to revolutions in physics? Chaos is present in the most of physical phenomena Role of chaos was not recognized before 20th century Only chaos can explain spontaneous creation of complicated structures Only chaos can produce order
Classical mechanics Newton s laws of motion Law of inertia Law of force F = m a Law of action and reaction Law of force is system of second order differential equations F i = m ẍ i, i = 1, 2, 3 Newton s law of gravitation F = κ m 1 m 2 r r 0 3 (r r 0 )
Example: motion in homogeneous grav. field equations of motion ẍ = 0, ÿ = g solution x(t) = x 0 + v 0x t, y(t) = y 0 + v 0y t 1 2 g t2 Exercise: Solve equations of motion using Maple
Example: harmonic oscillator Important model in physics, e.q. spring, atoms in lattice q = 0, equilibrium position m - mass of the point displacement q Displacement q Restoring force is proportional to the displacement F = k q, k is coefficient of rigidity m F - force
Equation of motion q + ω 2 q = 0, where ω = k m Solution can be written in several equivalent forms q(t) = A sin(ω t + φ 0 ) q(t) = C 1 sin ω t + C 2 cos ω t q(t) = α 1 e i ω t + α 2 e i ω t A amplitude, ω angular frequency, T = 2π/ω period ωt + φ 0 phase φ 0 initial phase Exercise: solve equation of motion in Maple
Energy of harmonic oscillator Work done by restoring force from amplitude to equil. position W = 0 A F dq = 1 2 k A2 = 1 2 m ω2 A 2 Energy of the oscillator is E = 1 2 m ω2 A 2 Potential of restoring force V (q) = 1 2 m ω2 q 2, so that F = V q, V (A) = E Kinetic energy T = 1 2 m q2 Exercise: show that E = T + V and E =constant
Phase diagram Eq. of motion q + ω 2 q is equivalent to first order system define momentum p = m q, then q = p m, ṗ = m ω2 q (Hamilton s equations) q and p are treated as independent variables Energy: E = p2 2m + 1 2 m ω2 q 2 Solution of eqs. of motion is a pair of functions q = q(t), p = p(t) Plane with coordinates (q, p) is phase space Graph of curve r(t) = (q(t), p(t)) is phase trajectory Phase diagram is set of phase trajectories Phase diagram can be obtained without solving eqs. of motion p = ± 2E q 2 for m = ω = 1 Exercise: plot phase diagram of harm. oscillator in Maple
Mathematical pendulum θ r m θ F N F t F g equilibrium position, θ = 0 Grav. force, F g = mg Tangential component, F t = m g sin θ Angular velocity, ω = θ Linear velocity, v = ω r Eq. of motion, F t = m v θ = g r sin θ
Equation of motion θ = ω 2 0 sin θ, where ω 0 = g r Introduce dimensionless variable d d t = ω 0 t, dt = ω 0 d t Equation of motion simplifies to θ + sin θ = 0 Solution is not an elementary function For θ 1 sin θ θ harmonic oscillator Numerical methods required
Phase diagram of math. pendulum Eq. of motion, θ + sin θ = 0 First integral E = 1 2 θ 2 cos θ Define momentum p = θ Eq. of phase trajectory p = 2E + 2 cos θ Exercise: plot phase diagram in Maple and discuss several regimes of pendulum
Dynamical systems System described by n variables Phase space R n Coordinates x = (x 1,... x n ) in phase space Forces F = (F 1,... F n ), F a = F a (x) for autonomous system = F a (x, t) for non-autonomous system F a Eqs. of motion ẋ a (t) = F a (x, t) or ẋ(t) = F (x, t) Necessary conditions for chaos n 3 F a are non-linear functions
Mathematical pendulum θ + sin θ = 0 Variables x = (θ, p) = (θ, θ) Phase space - R 2 [θ, p] Eqs. of motion θ = p ṗ = sin θ Forces F = (p, sin θ) p F F θ