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For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or ...
A less trivial example of a degenerate critical point is the origin of the monkey saddle. The index of a non-degenerate critical point of is the dimension of the largest subspace of the tangent space to at on which the Hessian is negative definite.
Otherwise it is non-degenerate, and called a Morse critical point of . The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4]
The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space.
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The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each.
Critical variables are defined, for example in thermodynamics, in terms of the values of variables at the critical point. On a PV diagram, the critical point is an inflection point . Thus: [ 1 ]
The main points of the proof can be made clearer by considering a one-dimensional system with a Lagrangian (, ˙, ¨).The Euler–Lagrange equation is ˙ + ¨ = Non-degeneracy of means that the canonical coordinates can be expressed in terms of the derivatives of and vice versa.