Physics · Mechanics · Year 2

Hamiltonian mechanics

How can the whole future of a swing fit on a single map?

The situation

A swing is flying in a children's playground. At every moment its state is completely described by two numbers: the deflection angle $θ$ and the momentum $p$ . If we take these two numbers as a point on a map, the whole future of the swing becomes a single line on the map. On the big board next to the swing is exactly this map: oscillations, full rotations and the boundary that separates them. Why does the “fluid” on this map never get compressed — and which quantities stay completely unchanged?

Topic

Hamiltonian mechanics and phase space. Phase portrait of a pendulum: conservation laws from symmetry

Key ideas

Phase space and Hamilton's equations
Liouville's theorem
Symmetry and conservation laws

Formulas

H (θ, p) = \frac{p^{2}}{2 m l^{2}} + m g l (1 - \cos θ)

The Hamiltonian of the swing is the total energy; $p = m l^{2} \dot{θ}$ is the canonical momentum (angular momentum).

\dot{θ} = \frac{\partial H}{\partial p}, \dot{p} = - \frac{\partial H}{\partial θ}

Hamilton's equations: the velocity of a point on the map (the blue and red arrows on the board).

\frac{d F}{d t} = {F, H} + \frac{\partial F}{\partial t}, {F, G} = \sum_{i} (\frac{\partial F}{\partial q_{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q_{i}})

Poisson brackets: if ${F, H} = 0$ and $\partial F / \partial t = 0$ , then $F$ is a constant of motion.

T = 4 \sqrt{\frac{l}{g}} K (k), k = \sin \frac{θ_{0}}{2}

The period depends on the amplitude: $K$ is the complete elliptic integral; at the separatrix ( $θ_{0} \to 180 °$ ) $T \to \infty$ .

J = \oint p d θ \approx \frac{2 π E}{ω} = const

The action variable is the area enclosed by the orbit. If a parameter changes slowly, it is conserved (adiabatic invariant).

Theory

Phase space and Hamilton's equations

In Hamiltonian mechanics the state is given by a generalised coordinate $q$ and the canonical momentum $p = \partial L / \partial \dot{q}$ . The Hamiltonian $H = p \dot{q} - L$ (here the total energy) gives two first-order Hamilton's equations and attaches a velocity vector to every point of the $(θ, p)$ phase plane. Trajectories do not cross: from one state only one future emerges. On the board, closed lines are oscillations and wavy lines are full rotations; they are separated by the separatrix $H = 2 m g l$ : this motion approaches the inverted upper position in an infinitely long time. $H$ does not depend explicitly on time (homogeneity of time), so energy is conserved and the point always stays on one line. Model: a swing with a rigid rod (like the 360° swing ride at an amusement park), a point mass $m = 30$ kg, no friction.

Principle of least action

The action $S = \int L d t$ assigns a single number to each path. Hamilton's principle: among paths with fixed end points, the true motion makes $S$ stationary, and over a short interval a minimum. In the “Hamilton's principle” section, $ε \sin (π t / τ)$ is added to the true path ( $τ = 0.4 T_{0}$ — shorter than half a period): on the graph $S (ε) - S (0) \geq 0$ , with the minimum exactly at $ε = 0$ . If we regard the action taken along the true path as a function of the end point, then $\partial S / \partial q = p$ and $\partial S / \partial t = - H$ . For a given energy, the Maupertuis principle applies: the abbreviated action $S_{0} = \int p d q$ is stationary on the true trajectory.

Liouville's theorem

Phase flow is incompressible: $\partial \dot{q} / \partial q + \partial \dot{p} / \partial p = \partial^{2} H / \partial q \partial p - \partial^{2} H / \partial p \partial q = 0$ . So even if a blob of initial states is stretched into a thin spiral, its area (in many dimensions — phase volume) does not change. Friction ( $\dot{p} = \dots - γ p$ ) makes the divergence $- γ$ : the area decreases as $e^{- γ t}$ . The numerical method also matters: the symplectic Störmer–Verlet (leapfrog) scheme preserves the area exactly, whereas the explicit Euler method changes it at every step by a factor of $1 + h^{2} ω_{0}^{2} \cos θ$ . In statistical physics an ensemble flows in phase space like an incompressible fluid: $d ρ / d t = 0$ — the basis of equilibrium distributions.

Poisson brackets and canonical transformations

For any quantity, $d F / d t = {F, H} + \partial F / \partial t$ ; the basic brackets are ${q, p} = 1$ , ${q, q} = {p, p} = 0$ . A change of variables that preserves them is called canonical: in the new variables Hamilton's equations have the same form. It is convenient to write it with a generating function: for a harmonic oscillator, the transformation $F_{1} = \frac{1}{2} m ω q^{2} \cot φ$ gives $q = \sqrt{2 I / (m ω)} \sin φ$ , $p = \sqrt{2 m ω I} \cos φ$ ( $I = J / 2 π$ ), and $H = ω I$ . The $(φ, J)$ view in “Adiabatic” mode is the exact version of this transformation for the pendulum; the simulation calculates the ${φ, J / 2 π}$ in the HUD numerically, and it comes out equal to 1.

Action–angle variables and the adiabatic invariant

In periodic motion the action $J = \oint p d q$ is the area enclosed by the orbit, and the angle $φ$ grows uniformly: $\dot{φ} = ω = \partial H / \partial I$ , $I = J / 2 π$ . If a parameter (the rope length) changes slowly compared with the period, both the energy and the frequency change, but $J$ stays almost constant — the adiabatic invariant. For small oscillations $J = 2 π E / ω$ : if the rope is shortened 2 times, both $ω$ and the energy increase $\sqrt{2}$ times (the work is done by the hand pulling the rope). If it is pulled quickly, the invariant is broken. The same idea works in magnetic traps: for a charged particle, $μ = m v_{⟂}^{2} / (2 B)$ is an adiabatic invariant, so a particle flying towards a strong field is reflected by a magnetic mirror — this is how the Earth's radiation belts are held (in detail — in the “Aurora and the Lorentz force” experiment).

Symmetry and conservation laws

To every continuous symmetry of the Lagrangian there corresponds a conserved quantity (Noether's theorem): homogeneity of time → energy, homogeneity of space → momentum, isotropy → angular momentum. If a coordinate does not enter $H$ , it is cyclic: ${\dot{p}}_{x} = - \partial H / \partial x = 0$ . In a circular bowl, $φ$ is cyclic, $p_{φ} = L_{z}$ is constant, and the Routhian reduces the problem to a single radial coordinate: $V_{e f} (r) = V (r) + L_{z}^{2} / (2 m r^{2})$ — that is what the graph shows. In a closed system, homogeneity means the centre of mass moves uniformly. The Hamilton–Jacobi equation $\partial S / \partial t + H (q, \partial S / \partial q) = 0$ separates in cyclic coordinates: $S = - E t + L_{z} φ + S_{r} (r)$ ; the Kepler problem is also solved this way in polar and spherical coordinates (in detail — in the “Kepler's laws” experiment). Model: the surface of the bowl is the graph of the potential $V = m g h (x, y)$ ; the puck moves on a horizontal plane without friction (slopes are taken to be small).

Did you know?

In 1911, at the first Solvay Congress, Hendrik Lorentz asked about a pendulum whose string is slowly shortened. Albert Einstein answered: if the length changes infinitely slowly, the ratio of the energy to the frequency, $E / ν$ , stays constant. This is one of the earliest precise examples of an adiabatic invariant.
Emmy Noether published her theorem linking symmetries and conservation laws in 1918, in the paper “Invariante Variationsprobleme”.

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.

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Free fall and air — Grade 10
Jet propulsion and the rocket — Grade 10
Viscosity and flow — Grade 10
Gyroscope and precession — Year 1
Lagrangian mechanics and chaos — Year 2