The Euler-Lagrange Equation and Conserved Momenta · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

From energy to motion

Science · Forces & Motion · ages 21-22
Name ______________________   Date ____________

  1. A coordinate never appears in L. What does that imply?

    • Its momentum is conserved
    • Its acceleration must be zero
    • The Lagrangian is invalid

  2. What does applying the Euler Lagrange equation to L return?

    • The constraint forces
    • The equation of motion for each coordinate
    • The numerical value of energy

  3. Which symmetry conserves angular momentum?

    • Shifting the clock
    • Rotation symmetry
    • Rescaling the mass

  4. The potential skips some direction. What follows?

    • The coordinate must be removed
    • A force appears along it
    • No force acts along it, so motion there coasts

  5. You apply the recipe to the pendulum L and get the Newton swing equation. What does that show?

    • The energy method is consistent with forces
    • The Lagrangian must be wrong
    • Cyclic coordinates cannot exist

  6. A cyclic coordinate always gives a conserved momentum, with no extra conditions.

    Circle one:   True   False

  7. L depends on positions but a friend claims energy cannot be conserved. What is wrong?

    • Time symmetry still guards energy when L skips time itself
    • Energy needs friction
    • Conservation forbids Lagrangians

  8. Could you get the pendulum equation from its Lagrangian and name the symmetry behind angular momentum?

    • Only by returning to free body diagrams
    • No, pendulums break the recipe
    • Yes: apply the recipe for the swing equation; rotation symmetry guards angular momentum
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Answer key

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

From energy to motion W1-mt_GSEFABbEXI-s1

  1. Its momentum is conserved · A missing coordinate is cyclic, so its conjugate momentum stays constant.
  2. The equation of motion for each coordinate · Each coordinate yields its own equation of motion through the fixed recipe.
  3. Rotation symmetry · Rotating the description changes nothing, so the corresponding momentum coasts.
  4. No force acts along it, so motion there coasts · No dependence means no push, which is the conserved momentum in disguise.
  5. The energy method is consistent with forces · Reproducing the known equation is the first consistency check on your L.
  6. True · Missing from L means the recipe leaves its conjugate momentum constant.
  7. Time symmetry still guards energy when L skips time itself · When L has no explicit time dependence, the clock symmetry yields energy conservation.
  8. Yes: apply the recipe for the swing equation; rotation symmetry guards angular momentum · The recipe gives the swing equation, and rotation symmetry explains the conserved quantity.
Worksheet · LightMySky