Hamilton's Principle and the Action · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The path that stands still

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

  1. What marks the physical path among all imagined ones?

    • Stationary action
    • Longest duration
    • Largest peak force

  2. What quantity does Hamilton principle concern?

    • The force at each instant
    • The momentum at the end
    • The action of a whole path

  3. In the fixed-endpoint variation behind Hamilton's principle, varying a path keeps both endpoints fixed.

    Circle one:   True   False

  4. Two arcs join the same fixed start and end positions at fixed times. Which property must the real arc have?

    • Stationary action
    • The highest peak
    • The shortest duration

  5. Stationarity under all pinned bends returns what?

    • Conservation of energy directly
    • The uncertainty relation
    • The Euler-Lagrange equation

  6. What does stationary mean, precisely?

    • The action value is exactly zero for the path
    • Zero first-order change under small bends
    • The path is a straight line

  7. You vary a path but let the end time slide. What follows?

    • Nothing changes; the interior equation follows exactly as before with no extra terms
    • Extra end conditions appear; the interior equation still follows
    • The action becomes exactly zero

  8. A student says the principle always picks least action. What is wrong?

    • Action has no minima at all
    • Stationary includes saddles, not only minima
    • Only instants matter in mechanics, never whole paths
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Answer key

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

The path that stands still W1-mt_a273REeeGu-s1

  1. Stationary action · Nature takes a path whose action stands still; several paths can be stationary at once.
  2. The action of a whole path · The principle judges entire paths by one number.
  3. True · Pinned ends define an admissible variation.
  4. Stationary action · With fixed endpoint positions and times, a real path must make the action stationary; several paths can satisfy this, so uniqueness is not assumed.
  5. The Euler-Lagrange equation · The variational condition is that equation.
  6. Zero first-order change under small bends · First-order stillness, not zero value.
  7. Extra end conditions appear; the interior equation still follows · Sliding the end time leaves boundary terms that must vanish, giving extra end conditions on the free end; bends vanishing near the ends still yield the interior Euler-Lagrange equation.
  8. Stationary includes saddles, not only minima · Stillness covers saddles too; for ordinary positive kinetic energy a saddle already qualifies, so least is too narrow a claim.
Worksheet · LightMySky