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Noether theorems volume, initial import (A-12)

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{{DISPLAYTITLE:Variational integrators and discrete symmetry}}

'''This page works its case,''' and the result is exact rather than approximate,
which is the reason the page exists.

== The problem with simulating mechanics ==

Solve a mechanical system numerically the obvious way — step the equations of
motion forward in small increments — and the conserved quantities drift. Energy
creeps up or down, an orbit spirals slowly in or out, and refining the step size
reduces the drift without removing it.

That is not a bug in the arithmetic. Nothing in the method knows the quantities
are supposed to be conserved.

== Discretise the action instead ==

The alternative is to discretise the object the theorems are about. Replace the
integral by a sum over steps, with a function <math>L_{d}</math> of consecutive
positions standing in for the integral across one step:

<math>S_{d}=\sum_{k}L_{d}\left(q_{k},q_{k+1}\right)\qquad\qquad\mathrm{(1)}</math>

Require the sum to be stationary under variations of the interior points. Each
<math>q_{k}</math> appears in two terms, so differentiating with respect to it
gives

<math>D_{2}L_{d}\left(q_{k-1},q_{k}\right)+D_{1}L_{d}\left(q_{k},q_{k+1}\right)=0\qquad\qquad\mathrm{(2)}</math>

where <math>D_{1}</math> and <math>D_{2}</math> denote differentiation with
respect to the first and second argument. This is the equation the simulation
steps forward.

== The conserved quantity, exactly ==

Now suppose <math>L_{d}</math> is invariant under a rigid shift of both its
arguments:

<math>L_{d}\left(q_{k}+\varepsilon,\ q_{k+1}+\varepsilon\right)=L_{d}\left(q_{k},q_{k+1}\right)\qquad\qquad\mathrm{(3)}</math>

Differentiate (3) with respect to <math>\varepsilon</math> at zero:

<math>D_{1}L_{d}+D_{2}L_{d}=0\qquad\qquad\mathrm{(4)}</math>

as an identity in both arguments. Define

<math>J_{k}=-D_{1}L_{d}\left(q_{k},q_{k+1}\right)\qquad\qquad\mathrm{(5)}</math>

By (4), <math>J_{k}=D_{2}L_{d}(q_{k},q_{k+1})</math>, and therefore also
<math>J_{k-1}=D_{2}L_{d}(q_{k-1},q_{k})</math>. But (2) says
<math>D_{2}L_{d}(q_{k-1},q_{k})=-D_{1}L_{d}(q_{k},q_{k+1})=J_{k}</math>. Hence

<math>J_{k-1}=J_{k}\qquad\qquad\mathrm{(6)}</math>

'''Exactly, at every step, for any step size.''' Not to first order, not to the
accuracy of the scheme. The quantity is the same number for ever, and it is the
discrete momentum.

== What the derivation used ==

Only (3) and (2). No limit was taken, no step size was assumed small, and no
property of the continuous theory was borrowed.

'''The theorem's structure did not need continuity.''' What it needed was an
action and a symmetry of it, and both survive discretisation intact. That is why
integrators built this way conserve momentum to machine precision while
conventional ones do not.

== The one that does not come free ==

Energy is different. Energy corresponds to translation in the independent
variable, and discretising fixes the step, which breaks that symmetry.

So discrete-action methods conserve momentum exactly and energy only
approximately — though they keep it bounded rather than letting it drift, for
reasons beyond this page. The pattern is exactly what the theorems predict:
'''the symmetries you keep are the quantities you conserve, and discretising is a
choice about which symmetries to keep.'''

''Provenance: (1) to (6) are standard in the numerical-analysis literature and are
derived above rather than cited. No source was consulted. See
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/Applications/Variational integrators and discrete symmetry - Revision history

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