Opened 4 years ago

Closed 4 years ago

#27548 closed enhancement (fixed)

Introduce periodic coordinates on manifolds

Reported by: Eric Gourgoulhon Owned by:
Priority: major Milestone: sage-8.8
Component: geometry Keywords: coordinate chart, manifold
Cc: Travis Scrimshaw Merged in:
Authors: Eric Gourgoulhon Reviewers: Travis Scrimshaw
Report Upstream: N/A Work issues:
Branch: 1718b1d (Commits, GitHub, GitLab) Commit: 1718b1d43dcc024c5bb1b9b1be6afa0142c99b36
Dependencies: Stopgaps:

Status badges

Description (last modified by Eric Gourgoulhon)

This ticket introduces periodic coordinates on manifolds for more flexibility. In particular, this fixes an issue with the azimuthal angle phi being limited to the interval (0,2*pi). For instance, the numerical computation of a geodesic that is winding around the center in Schwarzschild spacetime fails:

sage: M = Manifold(4, 'M', structure='Lorentzian')
sage: X.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: g = M.metric()
sage: g[0,0], g[1,1] = -(1-2/r), 1/(1-2/r)
sage: g[2,2], g[3,3] = r^2, (r*sin(th))^2
sage: p0 = M((0, 8, pi/2, pi))  # initial point
sage: v0 = M.tangent_space(p0)((1.297, 0, 0, 0.064))  # initial tangent vector
sage: s = var('s')
sage: geod = M.integrated_geodesic(g, (s, 0, 1500), v0)
sage: geod.solve() # numerical integration
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
...
ValueError: the 7th point (initial point being the '0th' point) of the 
numerical solution (obtained for a curve parameter equal to 105.0) is 
out of the chart domain

This is because phi reaches 2*pi at some point in the integration.

With the branch in this ticket, it suffices to declare phi to be a periodic coordinate, by adding the keyword periodic in the chart definition

sage: X.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):periodic:\phi')

to allow for the full integration of the geodesic.

The periodicity of the coordinate is taken into account in point comparisons:

sage: M((0, 8, pi/2, pi)) == M((0, 8, pi/2, 3*pi))
True
sage: M((0, 8, pi/2, pi)) == M((0, 8, pi/2, -pi))
True

Change History (6)

comment:1 Changed 4 years ago by Eric Gourgoulhon

Branch: public/manifolds/periodic_coords
Cc: Travis Scrimshaw added
Commit: 1d658e0e3f95a6c8389e7b0b36eea914078bfff8
Status: newneeds_review

New commits:

d27d464Add periodic coordinates in manifold charts
2e2543eTreatment of periodic coordinates in point comparison; add periodic azimuthal coordinate in Euclidean spaces
1d658e0Correct typo in error message in IntegratedCurve.solve()

comment:2 Changed 4 years ago by Eric Gourgoulhon

Description: modified (diff)

comment:3 Changed 4 years ago by git

Commit: 1d658e0e3f95a6c8389e7b0b36eea914078bfff81718b1d43dcc024c5bb1b9b1be6afa0142c99b36

Branch pushed to git repo; I updated commit sha1. New commits:

1718b1dFix two pyflakes errors in src/sage/manifolds/differentiable

comment:4 Changed 4 years ago by Travis Scrimshaw

Reviewers: Travis Scrimshaw
Status: needs_reviewpositive_review

LGTM.

comment:5 Changed 4 years ago by Eric Gourgoulhon

Thanks for the review!

comment:6 Changed 4 years ago by Volker Braun

Branch: public/manifolds/periodic_coords1718b1d43dcc024c5bb1b9b1be6afa0142c99b36
Resolution: fixed
Status: positive_reviewclosed
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