Physics · Mechanics · Year 2

Lagrangian mechanics and chaos

Why does a double pendulum released identically swing differently within a few seconds?

The situation

A double pendulum is mounted on the wall of a physics laboratory. The teacher releases it twice from exactly the same position: for the first few seconds the motion is identical, and then it is completely different. The laws are exact, the equations are known, there is no chance involved — and the difference between the two copies was too small to see. So why can't the motion be predicted after just a few seconds?

Topic

Lagrangian mechanics and chaos. Double pendulum: a small difference — completely different motion

Key ideas

Euler–Lagrange equations
Deterministic chaos
Lyapunov exponent

Formulas

L = \frac{1}{2} (m_{1} + m_{2}) l_{1}^{2} {\dot{θ}}_{1}^{2} + \frac{1}{2} m_{2} l_{2}^{2} {\dot{θ}}_{2}^{2} + m_{2} l_{1} l_{2} {\dot{θ}}_{1} {\dot{θ}}_{2} \cos (θ_{1} - θ_{2}) + (m_{1} + m_{2}) g l_{1} \cos θ_{1} + m_{2} g l_{2} \cos θ_{2}

The Lagrangian $L = T - V$ : the generalised coordinates are the two angles.

S = \int_{t_{1}}^{t_{2}} L d t, δ S = 0 ⟹ \frac{d}{d t} \frac{\partial L}{\partial {\dot{θ}}_{i}} - \frac{\partial L}{\partial θ_{i}} = 0

From the principle of least action (Hamilton's principle) — the Euler–Lagrange equations ( $i = 1, 2$ ).

(m_{1} + m_{2}) l_{1} {\ddot{θ}}_{1} + m_{2} l_{2} {\ddot{θ}}_{2} \cos Δ + m_{2} l_{2} {\dot{θ}}_{2}^{2} \sin Δ + (m_{1} + m_{2}) g \sin θ_{1} = 0

The first equation, $Δ = θ_{1} - θ_{2}$ .

l_{2} {\ddot{θ}}_{2} + l_{1} {\ddot{θ}}_{1} \cos Δ - l_{1} {\dot{θ}}_{1}^{2} \sin Δ + g \sin θ_{2} = 0

The second equation. Both are coupled and nonlinear.

| δ θ (t) | \approx | δ θ_{0} | e^{λ t}

In chaos a small difference grows exponentially; $λ$ is the Lyapunov exponent.

Theory

The Lagrangian method

Instead of forces, we write down the energy. The number of independent quantities that completely determine the state of a system is its number of degrees of freedom. Two point masses in a plane have 4 coordinates, and the two rods impose 2 conditions: $4 - 2 = 2$ . So the generalised coordinates are the two angles $θ_{1}, θ_{2}$ , and the generalised velocities are ${\dot{θ}}_{1}, {\dot{θ}}_{2}$ . The Lagrangian of a system of point masses is $L = T - U = \sum_{a} m_{a} v_{a}^{2} / 2 - U$ : if we write $v_{a}$ in terms of the angles, we get the formula above. The generalised forces $Q_{i} = - \partial U / \partial θ_{i}$ are, here, the moments of the gravitational forces (N·m). The forces in the rods do not enter the equations at all. The Euler–Lagrange equations give two coupled nonlinear equations — the simulator solves them by the RK4 method with a very small step. Watch $L$ , $Q_{i}$ and ${\dot{θ}}_{i}$ in the “Lagrangian quantities” section of the panel.

Constraints and d'Alembert's principle

The rods are holonomic constraints: they impose conditions on the coordinates ( $x_{1}^{2} + y_{1}^{2} = l_{1}^{2}$ and so on) and reduce the number of degrees of freedom. The reaction forces of the constraints (the rod forces $T_{1}, T_{2}$ ) are always perpendicular to the relative velocity of the bobs, so they do no work — their total power shown in the panel is always zero. For such ideal constraints there is d'Alembert's principle: the work of the active forces and the inertial “forces” $- m_{a} {\vec{a}}_{a}$ on any small displacement consistent with the constraint is zero; the Euler–Lagrange equations follow from it directly. If the reactions are needed, they are calculated from the motion found, using Newton's law — this is how $T_{1}, T_{2}$ in the panel are obtained (a negative value means the rod is compressed). Nonholonomic constraints impose conditions on the velocities, and they cannot be integrated into a condition between the coordinates (a ball rolling without slipping, a skate): they do not reduce the number of coordinates, and they enter the equations through Lagrange multipliers. If heavy rigid bodies are taken instead of rods, the method does not change: moments of inertia are written in place of $m l^{2}$ — a system of rigid bodies is also described by these same two angles.

Principle of least action

The action is the integral of the Lagrangian over time: $S = \int_{t_{1}}^{t_{2}} L d t$ (J·s). Hamilton's principle: given the initial and final states, of all the imaginable paths between them the system chooses the one along which the action is stationary ( $δ S = 0$ ); the calculus of variations turns this condition into exactly the Euler–Lagrange equations. The panel shows the accumulating $S$ for the main pendulum. For a free particle $L = m v^{2} / 2$ (Galilean invariance — see “Vectors and relative motion”), and for a relativistic free particle $L = - m c^{2} \sqrt{1 - v^{2} / c^{2}}$ . The Hamiltonian and phase space are in the “Hamiltonian mechanics” experiment.

Chaos: the “butterfly effect”

At large angles the motion is chaotic: the initial difference grows roughly as $e^{λ t}$ . On the lower graph $\ln | Δ θ |$ first rises as a straight line (its slope is $λ$ ) and then saturates. If you reduce $δ$ by a factor of 1000, the time to divergence increases by only $\ln 1000 / λ$ — a few seconds. That is why the motion is deterministic, yet cannot be predicted far into the future.

Small oscillations — normal modes

At small angles $\sin θ \approx θ$ , the equations are linear and the motion is the sum of two normal modes. For $m_{1} = m_{2}$ , $l_{1} = l_{2} = l$ we have $ω^{2} = \frac{g}{l} (2 \mp \sqrt{2})$ ; in the slow mode the bobs swing to the same side, $θ_{2} / θ_{1} = \sqrt{2}$ , and in the fast mode to opposite sides, $θ_{2} / θ_{1} = - \sqrt{2}$ . There is no chaos here — the copies stay together.

Numerical integration and the model

The computer calculates time in steps of $h$ . The simple Euler method makes an error at every step, and the energy keeps growing by itself; RK4 is fourth-order: the error is roughly proportional to $h^{4}$ . There is no friction, so the energy must stay constant — this is the best test of an integrator. The model: point masses, massless rigid rods, no air resistance.

Did you know?

In 1961 the meteorologist Edward Lorenz, when recalculating a weather model, entered the starting number 0.506127 rounded to 0.506. A few (model) months later the “weather” turned out completely different — this observation was the beginning of chaos theory.
The term “butterfly effect” comes from the title of Lorenz's 1972 talk: “Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?”

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Inclined plane and Atwood machine — Grade 10
Free fall and air — Grade 10
Jet propulsion and the rocket — Grade 10
Viscosity and flow — Grade 10
Gyroscope and precession — Year 1
Hamiltonian mechanics — Year 2