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Mathematical Physics

By John Earman

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If the family of solutions of the family of initial value problems y˙ = g(φ(y, ξ)), y(0) = 0, with parameter ξ ∈ R , is given by ρ : R × Rn → R, then ψ, defined by ψ(t, ξ) = φ(ρ(t, ξ), ξ) is the flow of the differential equation x˙ = g(x)f (x). n Proof. By definition ψ(0, ξ) ≡ ξ, and by the chain rule d ψ(t, ξ) = g(ψ(t, ξ))f (ψ(t, ξ)). 11) by a reparametrization of time. In the most important special cases the function g is constant. If its constant value is c > 0, then the reparametrization of the differential equation x˙ = cf (x) by ρ = ct results in the new differential equation dx = f (x).

8 Manifolds In this section we will define the concept of a manifold as a generalization of a linear subspace of Rn , and we will begin our discussion of the central role that manifolds play in the theory of differential equations. 34 1. Introduction to Ordinary Differential Equations Let us note that the fundamental definitions of calculus are local in nature. For example, the derivative of a function at a point is determined once we know the values of the function in some neighborhood of that point.

What are the properties of the flow of grad H? More generally, for a smooth function G : Rn → R (maybe n is odd), let us define the associated gradient system x˙ = grad G(x). Because a conservative force is the negative gradient of a potential, many authors define the gradient system with potential G to be x˙ = − grad G(x). The choice of sign simply determines the direction of the flow. Prove the following statements: (a) A gradient system has no periodic orbits. (b) If a gradient system has a rest point, then all of the eigenvalues of its linearization at the rest point are real.

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