Eigenvalues of the Frenet formulas and angular velocity

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SUMMARY

The discussion centers on the eigenvalues of the curvature matrix K derived from the Frenet-Serret formulas for a circular helix defined by the parametric equations \(\vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt)\). It is established that the nonzero eigenvalue of \(K^2\) is \(-\alpha^2\). Participants clarify that this conclusion can be drawn from the properties of the curvature matrix without direct computation, leveraging theoretical insights from differential geometry.

PREREQUISITES
  • Understanding of Frenet-Serret formulas
  • Knowledge of eigenvalues and eigenvectors in linear algebra
  • Familiarity with parametric equations of curves
  • Basic concepts of differential geometry
NEXT STEPS
  • Study the derivation of the Frenet-Serret formulas in detail
  • Learn how to compute eigenvalues of matrices, specifically for curvature matrices
  • Explore the geometric interpretation of eigenvalues in the context of curves
  • Investigate the properties of circular helices and their applications in physics
USEFUL FOR

Mathematicians, physicists, and students studying differential geometry or vector calculus, particularly those interested in the properties of curves and their curvature matrices.

ForMyThunder
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So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand that you can compute the eigenvalues of the matrix to verify this but how can you say this without computation?
 
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ForMyThunder said:
So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand that you can compute the eigenvalues of the matrix to verify this but how can you say this without computation?

Can you show me why the eigen value is -alpha^2?

I get +-alpha^2/(a.alpha^2 + b^2)
 

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