We began by saying that some things never change, and that knowing which ones is half of physics. We asked three questions: what conservation laws can decide on their own, how small steps add up, and how to tame many bodies. Here is what the chapter’s eight questions showed.
1. What can conservation laws decide on their own?
For two bodies colliding elastically in a line, momentum and energy determine the outcome completely, and that single result was used again and again. For more bodies, they do not. In Section 10.1, a row of marbles had many outcomes consistent with both laws, and it took a further physical fact (marbles cannot pass through each other) to rule some out. The square of the momentum minus the energy gave a sum of cross terms that had to vanish, an inequality argument that proved certain motions impossible. And the familiar behaviour of Newton’s cradle turned out to depend on details beyond the conservation laws: how the shock travels through the balls.
Conservation laws also exposed hidden assumptions. In Section 10.3, firing guns all at once gave more speed than firing them one at a time, but only because it used more energy; holding the right quantity fixed reversed the conclusion.
2. How do small steps add up?
Several sections repeated one simple step. In Section 10.2, a collision that can at most double a speed became, through a chain of lighter and lighter balls, unlimited gain: the principle behind the basketball and tennis ball and behind gravitational slingshots. In Section 10.3, many small firings became an integral, and Tsiolkovsky’s logarithm, which explains why rockets are mostly fuel and why they have stages. In Section 10.4, each bounce off a moving piston added \(2u\) to a molecule’s speed, and an exact recurrence revealed that speed times length stays almost constant: an adiabatic invariant, and the reason compressed gases heat up. And in Section 10.5, infinitely many bounces took a finite time, because the times formed a convergent geometric series.
3. How do we tame many bodies?
By choosing coordinates and exploiting symmetry. In Section 10.6, the centre of mass moved uniformly and the separation behaved like a single particle with the reduced mass; a hidden plane even explained why the separation squared was a quadratic in time. In Section 10.7, symmetry and integration showed that a spherical shell exerts no gravity inside and acts as a point mass outside, which let a collapsing cloud of dust be treated as a single equation, with a free-fall time that sets the pace of star formation. And in Section 10.8, a symmetric guess solved a special case of the unsolvable three-body problem: Lagrange’s rotating triangle, realised in the Trojan asteroids.
4. A toolkit for mechanics
| Method | Where you used it |
|---|---|
| Conservation laws as constraints | Newton’s cradle (10.1) |
| Change of frame | Collisions (10.2); piston (10.4) |
| Iterating one result | Stacked balls (10.2) |
| Momentum for variable mass | Rockets (10.3) |
| Sum to integral | Rocket equation (10.3); shell (10.7) |
| Recurrence and telescoping | Piston (10.4) |
| Geometric series | Wedge (10.5) |
| Centre of mass and reduced mass | Two bodies (10.6) |
| Energy integral \(v\,\mathrm{d}v/\mathrm{d}z\) | Two bodies (10.6); collapse (10.7) |
| Symmetric guess | Three stars (10.8) |
Where next
Chapter 11 turns from conservation to oscillation and rotation: resonance, normal modes, the real pendulum, and motion in accelerating and rotating frames.
The energy method of Section 8.2 and the limit cycles of Section 8.4 are the differential-equation side of the same ideas.
The chaos Poincaré found in the three-body problem is the subject of Section 8.7.
Daniel Kleppner & Robert Kolenkow, An Introduction to Mechanics: the classic first-year text; demanding but rewarding.
David Morin, Introduction to Classical Mechanics: hundreds of problems, many of them as surprising as the ones in this chapter.
Richard Feynman, The Character of Physical Law: lectures for a general audience on conservation laws, symmetry and gravity.
Dwight Neuenschwander, Emmy Noether’s Wonderful Theorem: why symmetry implies conservation.
Feynman said that there is no known exception to the conservation of energy. This chapter has tried to show how much they can do: rule out impossible outcomes, reveal hidden invariants, and turn hopelessly complicated systems into ones we can understand. But it has also shown their limits. They constrain, but they do not always decide; and where they stop, the details of forces, symmetries and frames must take over.