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>
40 lines
1.9 KiB
Python
40 lines
1.9 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,41):
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for y0 in np.linspace(-box,box,41):
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x=np.array([x0,y0])
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for _ in range(100):
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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>0.1: dx*=0.1/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.hypot(*x)<0.02: continue
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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-4 for p in pts):
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ev=np.linalg.eigvalsh(H)
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pts.append((x,'min' if ev.min()>0 else ('max' if ev.max()<0 else 'saddle')))
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return pts
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np.seterr(all='ignore')
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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); w2,_=omega2('log',X,C,Rc,0)
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pts=stationary(X,C,0.0,2.5)
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print(f"inertial mu={mu}: {len(pts)} off-centre stationary points:",[(round(np.hypot(*x),4),round(np.degrees(np.arctan2(x[1],x[0]))%360,1),k) for x,k in pts])
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for i in (0,3):
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others=[j for j in range(len(X)) if j!=i]
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g,H=grad_hess(X[others],C[others],w2,X[i]); ev=np.linalg.eigvalsh(H)
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gi,Hi=grad_hess(X[others],C[others],0.0,X[i])
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print(f" body {i} (c={C[i]}): rotating-frame field of the others: |grad|={np.linalg.norm(g):.1e}, Hessian eig={ev.round(4)} -> {'min' if ev.min()>0 else ('max' if ev.max()<0 else 'saddle')}; inertial |grad|={np.linalg.norm(gi):.4f} (= w2*r={w2*np.hypot(*X[i]):.4f})")
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