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Why is momentum conserved in a collision but not when a ball bounces off the floor? Emmy Noether's theorem gives a deeper answer, and a quick test to run before choosing a conservation law.
Who was Emmy Noether?
Noether was born on 23 March 1882 in Erlangen, Germany, and died on 14 April 1935 at Bryn Mawr, Pennsylvania (MacTutor biography, University of St Andrews). As a young woman she could at first only sit in on university courses, with each professor's permission. After women were allowed to matriculate, she earned a doctorate at Erlangen in 1907.
In 1915 David Hilbert and Felix Klein invited her to Göttingen, where Hilbert was working on ideas in relativity and wanted an expert in invariant theory. It took a long battle with the university before she was allowed the habilitation needed to teach, so Hilbert advertised her courses under his own name; a catalogue entry for the winter semester of 1916-17 read "Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4-6, no tuition." Permission finally came in 1919. In April 1933 the Nazis caused her dismissal from Göttingen because she was Jewish, and later that year she moved to Bryn Mawr College in the United States.
What does the theorem say?
Her 1918 paper, "Invariante Variationsprobleme", appeared in the Göttingen Nachrichten (pages 235 to 257) and combines the calculus of variations with Lie's theory of continuous groups (English translation by M. A. Tavel, arXiv). MacTutor quotes the mathematician B. L. van der Waerden's summary of one result: "To every infinitesimal transformation of the Lorentz group there corresponds a Conservation Theorem." According to MacTutor, Einstein praised the work in a letter to Hilbert, referring to Noether's penetrating mathematical thinking.
In everyday terms, the theorem says that if the laws governing a system stay the same under a smooth change, some quantity is conserved (Noether's theorem overview with textbook citations):
Same laws at every moment (shift in time) gives conservation of energy.
Same laws at every place (shift in space) gives conservation of linear momentum.
Same laws in every direction (rotation) gives conservation of angular momentum.
The symmetry is in the laws, not the object. A jagged asteroid tumbling in space still conserves angular momentum, because the laws of its motion are the same in every direction.
The symmetry test for JEE mechanics
Before writing a conservation equation, draw the system and ask three questions.
If I slide the whole system sideways, does anything outside it notice? If nothing external depends on position along that direction, momentum along it is conserved.
If I rotate the whole system about a point, does anything outside it notice? If not, angular momentum about that point is conserved.
Does anything outside the system change with time? If not, and there is no friction, total mechanical energy is conserved.
The link to forces is direct: a potential energy that does not change as you shift along x means no force along x, so ΣF_x = 0 and p_x stays constant by Newton's second law.
Worked example 1: block on a smooth wedge
A block slides down a smooth wedge that rests on a smooth floor. Take the block and wedge as the system.
Slide everything horizontally: the floor is the same everywhere and gravity is vertical, so nothing notices. Horizontal momentum is conserved.
Slide everything vertically: gravity and the floor do notice. Vertical momentum is not conserved: as the block speeds up downwards, gravity and the floor's push on the system do not balance.
Nothing changes with time and every surface is smooth: mechanical energy is conserved.
Worked example 2: a planet and the Sun
Treat the Sun as fixed. Its pull on the planet always points to the Sun's centre. Rotate the orbit about that centre and the force law looks identical, so angular momentum about the Sun is conserved. About other points it generally is not. This is Kepler's equal-areas law in another form.
Worked example 3: ball bouncing on the floor
Shift the ball up and the floor notices: vertical momentum is not conserved during the bounce. Shift it sideways along a smooth floor and nothing notices: horizontal momentum is conserved.
Why students get conservation laws wrong
Most mistakes come from applying a law about the wrong point, along the wrong axis, or to the wrong system. Our guides to rotational motion mistakes and JEE mechanics mistakes list the common ones. The symmetry test catches many of them because it forces you to name the system and the direction first. For the momentum form of the second law, see Newton's original three laws.
If you want to check a problem where you are unsure which law applies, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Take the last three mechanics problems you got wrong and run the three-question test on each one.
