Three quantities that do not move while everything else does

At its defaults it draws three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

noether-charge is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

The same orbit in a potential multiplied by (1 + ε cos 2θ) with ε = 0.06, which makes the attraction ever so slightly stronger along one axis than the other. Nothing else is changed. The upper panel is the angular momentum along the trajectory and the lower panel is the energy, both read off the integrated motion. The angular momentum now swings by 2.814e-1 — it is not conserved, because the potential mentions the direction and so rotating the whole problem no longer gives back the same problem. The energy over the same run moves by 6.78e-8, which is the integrator's own noise at this step, because the potential still does not mention the time. That contrast is the argument: a numerical error would spoil both, and only the charge whose symmetry was removed has moved. The rate at which it moves is not free either — it is the torque −∂V/∂θ, and the accumulated torque tracks the whole history to 3.2e-9, although one is read off the integrated motion and the other is a quadrature of an expression in the position.

The angular momentum's excursion over one orbit against the strength of the term that breaks the rotational symmetry, both on logarithmic axes, for ε from 0.001 to 0.03. The points lie on a straight line of slope 1.0059, measured by least squares and not imposed — so halving the asymmetry halves the leak, exactly. That is the quantitative form of the theorem and the more useful one in practice: a symmetry that holds approximately gives a quantity that is conserved approximately, with the error first order in the breaking. Almost every conservation law used in physics is of this kind rather than the exact kind — momentum in a laboratory sitting on a planet, angular momentum in a galaxy that is not quite round — and the reason they remain useful is the slope on this chart. What the chart cannot show is where the line stops: at large enough ε the orbit stops being a perturbed ellipse and the excursion saturates at the whole of the angular momentum, which no power law describes.

Two bodies of unequal mass attracting one another, integrated from rest-frame-free initial conditions with a net momentum, so that both wander over the page. Neither path is straight and neither has a constant speed. The centre of mass, drawn through them, is straight to 3.5e-15 of the frame while the separation between the bodies changes by 1.94 over the same run. That straightness is a conservation law and it is the one nobody counts: the laws are the same in a frame moving at constant velocity, and the charge that symmetry supplies is Pt − MX, the total momentum times the time minus the total mass times the centre of mass. It is the only charge in ordinary mechanics that mentions the time explicitly, which is why it is usually stated as a sentence about the centre of mass rather than listed beside energy and momentum. Setting it constant and differentiating recovers the statement that the centre of mass moves uniformly — and, read the other way, the fact that two colliding bodies cannot move their common centre is the same symmetry, which is why a rocket needs exhaust.

physicscheck asserts something about noether-charge that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.