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

  3. Varying a path keeps both endpoints fixed.

    Circle one:   True   False

  4. Two arcs join the same throws; only one has stationary action. Which is real?

    • The stationary one
    • The higher one
    • The shorter-time one

  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 is exactly zero
    • No first-order change under small bends
    • The path is a straight line

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

    • Nothing, the principle ignores ends
    • Boundary terms appear and the derivation fails
    • 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 maxima and saddles, not only minima
    • Only instants matter, never wholes
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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 the path whose action stands still.
  2. The action of the whole path · The principle judges entire paths by one number.
  3. True · Pinned ends define an admissible variation.
  4. The stationary one · Stationarity is the selection rule.
  5. The Euler-Lagrange equation · The variational condition is that equation.
  6. No first-order change under small bends · First-order stillness, not zero value.
  7. Boundary terms appear and the derivation fails · Pinning ends is what kills the boundary terms.
  8. Stationary includes maxima and saddles, not only minima · Stillness covers more shapes than minima.
Worksheet · LightMySky