Questions: Phase Space Trajectories

Question 1

A harmonic oscillator starts with amplitude A = 2 m. A second identical oscillator starts with amplitude A = 4 m. How do their phase-space trajectories compare?

• A. They trace the same ellipse — amplitude does not affect the shape, only the period
• B. They trace concentric ellipses — the larger amplitude produces a larger ellipse scaled up from the origin
• C. They trace the same ellipse at different speeds — the larger amplitude oscillator moves faster around the ellipse
• D. The 4m oscillator traces a circle; only small-amplitude oscillators trace ellipses

Answer: (B)

Each energy level (and thus each amplitude) corresponds to a distinct ellipse in phase space: (x/A)^2 + (v/Aω)^2 = 1. Different amplitudes give different semi-axes, producing a family of concentric ellipses nested around the origin. The origin itself is a fixed point (equilibrium). This nesting of ellipses by energy level is a key feature of the phase portrait — it immediately shows that all oscillations are periodic (closed curves) and that amplitude is continuously adjustable without qualitative change in behavior.

Question 2

A damped harmonic oscillator (with friction) is plotted in phase space. What does its trajectory look like, and what does this reveal about the dynamics?

• A. A closed ellipse — damping only affects the period, not the overall shape
• B. A larger ellipse than the undamped case — damping adds energy to the system
• C. An inward spiral toward the origin — each cycle loses energy, reducing amplitude until the system comes to rest
• D. A straight line toward the origin — damped systems move directly to equilibrium without oscillating

Answer: (C)

Damping dissipates energy, so the amplitude decreases each cycle. In phase space, this means the trajectory spirals inward: each loop is a smaller ellipse than the previous one, converging toward the origin (equilibrium). The spiral structure immediately reveals that energy is being lost (motion is not periodic) and that the system approaches equilibrium asymptotically. The topology of the curve — open spiral vs. closed ellipse — distinguishes undamped from damped behavior at a glance, without solving any equations.

Question 3

Two different initial conditions for the same frictionless harmonic oscillator can share the same phase-space trajectory.

• T. True
• F. False

Answer: False

For a frictionless harmonic oscillator, the phase-space trajectory is an ellipse determined solely by total energy E. Two initial conditions with the same energy trace the same ellipse but start at different points on it. Two initial conditions with different energies trace different, non-intersecting ellipses. In general, phase-space trajectories for a deterministic system cannot cross — if two trajectories shared a point, the system would have two different futures from the same state, violating determinism. This non-crossing property is one of phase space's most important structural features.

Question 4

A closed curve in phase space indicates that the system's motion is periodic — it repeats the same sequence of states indefinitely.

• T. True
• F. False

Answer: True

A closed curve means the representative point eventually returns to its starting position in phase space — that is, the same (x, v) pair recurs. Since the state completely specifies the system's future evolution, returning to the same state means the subsequent motion is identical. This is exactly the definition of periodic motion. Conversely, open curves (spirals, hyperbolas, unbounded paths) represent non-periodic behavior: the system never returns to its initial state. The topology of phase-space curves — closed vs. open — directly encodes whether motion is periodic.

Question 5

What does it mean to say 'dynamics is geometry,' and why is the phase-space representation more powerful than plotting position or velocity against time separately?

Model answer: Saying 'dynamics is geometry' means that the qualitative behavior of a physical system — whether it oscillates, decays, is stable, or becomes chaotic — is encoded in the geometric structure of its phase-space trajectories, without needing to solve differential equations. Plotting position and velocity separately shows each variable's time history, but phase space shows the relationship between them as a single curve whose shape immediately reveals the system's global behavior: closed ellipse = periodic, inward spiral = damped, open curve = unbounded. Fixed points, stability, and qualitative changes in behavior become visible geometric features.

The time-series plots of x(t) and v(t) both require tracking time explicitly; phase space eliminates time to reveal the state-space structure directly. For the harmonic oscillator, the ellipse encapsulates all future and past behavior in a single geometric object. For more complex systems — nonlinear pendulums, chaotic systems — the phase portrait (the collection of all trajectories) reveals basins of attraction, separatrices, and strange attractors that time-series plots cannot show. This geometric viewpoint is the conceptual foundation of Hamiltonian mechanics and modern dynamical systems theory.