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Dynamics of the Heisenberg model and a theorem on stability
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48.It is also well-known that all spheres of non-zero radii are diffeomorphic and that, since , there is a unique differentiable structure on S2.
49.Non-Nöether symmetries are automorphisms which preserve the equations of motion but not the 2-form. Both Nöether and non-Nöether symmetries are associated with constants of motion.
50.For a differentiable function such that is a manifold, the vector δu is tangent to at u iff Φ(u + δu) = Φ(u) to first order in |δu|, or, , or, δu is a zero eigenvector of the Jacobian matrix at u. If now we consider the general dynamical system and linearize, we get . The Jacobian in this expression is called the stability matrix and is the set of the FPs of the flow. The vector δu is tangent to at the point u iff δu is a FP of the linear flow at u.
51.This orthogonality is the generalization of the concept of two planes in intersecting at right angles. Two subspaces V1 and V2 of an inner-product vector space V intersect orthogonally iff there exist 3 pairwise orthogonal subspaces of V, say W, U1, U2, such that V1 = W⊕U1 and V2 = W⊕U2. Equivalently, the orthogonal complements of V1∩V2 in V1 and V2 are orthogonal subspaces, i.e., (V1∩V2)⊥∩V1⊥(V1∩V2)⊥∩V2. Equivalently, (or .
52.The term “neighborhood of a point (set)” below denotes an open set which contains the point (set).
53.The condition ψ(g, 0) = 0 for all g ∈ P anticipates the application of the theorem to the HM. We could replace this condition with the assumption that ψ and F are continuous at t = 0 and make a small adjustment to the proof.
54.Note that for a Hamiltonian system, the existence of an eigenvalue in the open left half-plane implies the existence of a conjugate eigenvalue in the right half-plane and vice-versa. This is due to the fact that in a conservative flow there are no stable or unstable nodes, but just centres and saddles. Clearly, if there is any hope that a FP is stable, all eigenvalues of the stability matrix at that point must lie on the imaginary axis.
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