2.2 The Morse Lemma and its implications

The Morse Lemma is a game-changer for understanding smooth functions near critical points. It gives us a simple quadratic form that captures the function's behavior, making it easier to visualize and analyze.

This lemma helps classify critical points based on their index, which counts negative eigenvalues of the Hessian matrix. It's crucial for identifying minima, maxima, and saddle points, shaping our understanding of a function's topology.

Morse Lemma and Local Structure

Quadratic Approximation and Normal Form

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• Morse Lemma states that near a non-degenerate critical point $p$ of a smooth function $f$, there exist local coordinates $(y_1, \ldots, y_n)$ in which $f$ takes the form $f(y_1, \ldots, y_n) = f(p) - y_1^2 - \ldots - y_k^2 + y_{k+1}^2 + \ldots + y_n^2$
• Local coordinates $(y_1, \ldots, y_n)$ are chosen such that the critical point $p$ corresponds to the origin and the coordinate axes align with the principal directions of the Hessian matrix of $f$ at $p$
• Quadratic form $- y_1^2 - \ldots - y_k^2 + y_{k+1}^2 + \ldots + y_n^2$ in the Morse Lemma approximates the behavior of $f$ near the critical point $p$ up to second order
• Number of minus signs $k$ in the quadratic form is equal to the index of the critical point $p$, which is the number of negative eigenvalues of the Hessian matrix at $p$

Local Structure and Diffeomorphism

• Morse Lemma implies that near a non-degenerate critical point, the level sets of $f$ are diffeomorphic to quadratic hypersurfaces
• Diffeomorphism is a smooth invertible map with a smooth inverse, preserving the topological structure of the level sets
• Normal form of the Morse Lemma $f(p) - y_1^2 - \ldots - y_k^2 + y_{k+1}^2 + \ldots + y_n^2$ provides a canonical local representation of the function $f$ near the critical point $p$
• Local structure of $f$ near a non-degenerate critical point is completely determined by the index $k$, which classifies the critical point into different types (minimum, saddle, maximum)

Classification of Critical Points

Morse Functions and Non-Degenerate Critical Points

• Morse function is a smooth function whose critical points are all non-degenerate, meaning the Hessian matrix at each critical point is non-singular
• Non-degenerate critical points are isolated and have a well-defined index, which is the number of negative eigenvalues of the Hessian matrix
• Morse functions are generic in the space of smooth functions, implying that most functions encountered in practice are Morse functions
• Morse Lemma applies to non-degenerate critical points, providing a local quadratic approximation and normal form near each critical point

Index and Stability of Critical Points

• Index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point, ranging from $0$ to $n$ for an $n$-dimensional domain
• Critical points are classified based on their index:
  • Index $0$: local minimum, all eigenvalues are positive (stable)
  • Index $1$ to $n-1$: saddle point, mixture of positive and negative eigenvalues (unstable)
  • Index $n$: local maximum, all eigenvalues are negative (stable)
• Stability of critical points refers to their behavior under small perturbations of the function $f$
  • Minima and maxima are stable critical points, as small perturbations do not change their qualitative behavior
  • Saddle points are unstable, as small perturbations can alter the number and type of critical points in their vicinity