The Hamiltonian, Phase Space and the Canonical Equations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Trading velocity for momentum

Science · Forces & Motion · ages 22-23
Name ______________________   Date ____________

  1. For L = 2v² − 9x², what is the conjugate momentum p = ∂L/∂v?

    • 4v
    • 2v
    • −9x

  2. A cart obeys 2ẍ = −8x, with L = v² − 4x² and p = 2v. Which canonical pair follows?

    • ẋ = 2p and ṗ = −8x
    • ẋ = p/2 and ṗ = −8x
    • ẋ = p/2 and ṗ = −4x

  3. The Lagrangian is L equals one half m v squared minus one half k x squared, with v equals x prime. The conjugate momentum is p equals dL/dv. What is p?

    • m v
    • -k x
    • m v minus k x

  4. Hamilton's equations hold, and the Hamiltonian H(x, p) carries no explicit time dependence. Its value stays constant along the motion.

    Circle one:   True   False

  5. For the spring oscillator with H = p²/(2m) + ½kx² and m > 0, the state point sits where x > 0 and p > 0. Which way does x move next?

    • Leftward, since x is shrinking
    • Rightward, since x is growing
    • It stays put

  6. For the oscillator with H = p²/2 + x² (m = 1, k = 2), with x horizontal and p vertical, the state sits at x = 1, p = 1. Which way does it move?

    • Up and to the left
    • Down and to the right
    • Straight down, with no horizontal motion

  7. For H = p²/4 + 4x² with m = 2, k = 8, the energy is E = 16. What is the x-extent of the phase ellipse?

    • x = ±4
    • x = ±8
    • x = ±2

  8. For H equals p squared over 2m plus one half k x squared, the rule x prime equals dH/dp gives which result?

    • minus k x
    • k x
    • p over m

  9. Two spring carts with the same m and k are drawn on one phase plot with the same axes. One ellipse sits fully inside the other. What can you say about their energies?

    • The inner loop carries the larger energy
    • The inner loop has the smaller energy
    • Both loops must carry equal energy

  10. A particle has mass 4 kg and velocity 3 m/s, and its conjugate momentum is p = m v. Type the value of p in kg·m/s.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Trading velocity for momentum W1-mt_9bMDXQiq_n-s1

  1. 4v · Differentiating 2v² with respect to v gives 4v; the −9x² term has no v and contributes nothing.
  2. ẋ = p/2 and ṗ = −8x · p = 2v inverts to v = p/2, and H = p v − L = p²/4 + 4x², so ẋ = ∂H/∂p = p/2 and ṗ = −∂H/∂x = −8x.
  3. m v · Only the velocity term responds to dL/dv, and its derivative is m v.
  4. True · With no stray t inside H, Hamilton's equations give dH/dt = 0, so H is conserved; it equals T + V for ordinary time-independent coordinates with a natural Lagrangian.
  5. Rightward, since x is growing · Positive p makes ẋ = p/m positive, so x grows and the point drifts rightward; the vertical motion is a separate question.
  6. Down and to the right · ẋ = ∂H/∂p = p = 1 > 0 and ṗ = −∂H/∂x = −2x = −2 < 0, so the flow is down and to the right.
  7. x = ±2 · At p = 0 the condition 4x² = 16 gives x² = 4, so x = ±2.
  8. p over m · Differentiate with respect to p: the x term drops out and p squared over 2m gives p over m.
  9. The inner loop has the smaller energy · With the same m, k, and axes, a bigger loop means a larger fixed value E, so the inner loop has the smaller energy.
  10. 12 · Multiply mass by velocity: 4 kg times 3 m/s is 12 kg·m/s.
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