c981537904
Eyes facts for aetheria-gravity: regular and twisted alternating polygon equilibria (log closed form, Newtonian sources), linear stability (Muscas-Pasca-Stoica log thresholds reproduced, Klemperer, Thomson), the soft core's reach, PeriodAround's ring rule against the twisted equilibrium spin, and rotating-frame stationary points against the drawn potential. Solver scripts beside the note. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
39 lines
1.7 KiB
Python
39 lines
1.7 KiB
Python
import numpy as np
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from rosette import *
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def grad_hess(X,C,w2,x):
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g=np.zeros(2); H=np.zeros((2,2))
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for j in range(len(X)):
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d=x-X[j]; s2=d@d
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g+=C[j]*d/s2; H+=C[j]*(np.eye(2)/s2-2*np.outer(d,d)/s2**2)
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g-=w2*x; H-=w2*np.eye(2)
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return g,H
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def stationary(X,C,w2,box):
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pts=[]
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for x0 in np.linspace(-box,box,61):
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for y0 in np.linspace(-box,box,61):
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x=np.array([x0,y0])
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for _ in range(60):
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g,H=grad_hess(X,C,w2,x)
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try: dx=np.linalg.solve(H,-g)
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except: break
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st=np.linalg.norm(dx)
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if st>0.2: dx*=0.2/st
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x=x+dx
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if np.linalg.norm(dx)<1e-13: break
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g,H=grad_hess(X,C,w2,x)
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if np.linalg.norm(g)<1e-9 and min(np.hypot(*(x-X[j])) for j in range(len(X)))>1e-3 and np.hypot(*x)<box*1.5:
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if all(np.hypot(*(x-p[0]))>1e-6 for p in pts):
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ev=np.linalg.eigvalsh(H)
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# potential phi = sum c ln s - w2 r^2/2 (a well is -inf at bodies: phi is the gravitational potential, force = -grad phi)
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kind='min' if ev.min()>0 else ('max' if ev.max()<0 else 'saddle')
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pts.append((x,kind))
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return pts
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for mu in (1.0,0.7):
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r,_=closed_log_twisted(3,mu)
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X,C,Rc=config(3,1,mu,r,np.pi/3)
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w2,_=omega2('log',X,C,Rc,0)
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for label,ww in (('rotating effective',w2),('inertial (drawn)',0.0)):
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pts=stationary(X,C,ww,2.5)
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print(f"== hexagon mu={mu} rho={r:.4f} {label}: {len(pts)} stationary points")
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for x,k in sorted(pts,key=lambda p:(np.hypot(*p[0]),np.arctan2(p[0][1],p[0][0]))):
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print(f" r={np.hypot(*x):.4f} theta={np.degrees(np.arctan2(x[1],x[0]))%360:7.2f} {k}")
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