● Domain 01 · Physics
Symmetry that cannot slip.
Noether ties every continuous symmetry to a conserved current. Hold the symmetry approximately and you hold the law approximately.
Toroidal solenoidA winding that closes on itself. Ampere’s law traps the field inside a toroid and lets it out of a straight solenoid, and below the transition the metal expels it altogether.Meissner and Ochsenfeld (1933) · flux quantisation, London (1950); Deaver & Fairbank, PRL 7 (1961)
The failure is quiet
A symmetry placed in the objective holds only where the optimiser saw data. Away from it the symmetry slips, the conserved current goes with it, and nothing in the output says so.
What changes
Put the group action in the construction and the symmetry is exact for every input. The conserved current follows from the object, and can be checked on the weights.
Demonstration
One potential, two integrations.
Fitted integration
A fitted force law is not exactly central, so it turns the conserved vector, and not exactly inverse square, so the conic opens. The faint line is the encoded column.
Encoded integration
The kick is exactly along the line to the centre, so nothing in the update can torque the angular momentum. The conic closes and the plane holds.
The road the body has taken, brightest where it is now.
The angular momentum. Its length fixes the conic, its direction fixes the plane.
The plane the orbit began in, so a tip out of it is visible.
How far the vector has left its start. The dashed line is where it began.
Angular momentum is carried by the construction here rather than learned, so conservation is a property of the update itself. It is a vector, so both its length and its direction are measured. Both columns start from the same condition, in the same potential, at the same step, under the same integrator, and differ by one term. The fitted column loses nine per cent of the momentum and tips the plane twenty degrees; the encoded column holds both at double precision, and holds them at every step size the figure runs.
Sources
∂μ jμ = 0
Noether's theorem · Noether, Nachr. Ges. Wiss. Göttingen (1918)
{H, L} = 0 ⇒ dL/dt = 0
Conserved quantity from a Poisson-commuting generator · Arnold, Mathematical Methods of Classical Mechanics (1978)
Is this your problem?
Tell us what has to hold, and what happens if it does not.
xzenoverseindustries@gmail.com · answered within two business days