Near an unstable equilibrium of a mechanical system

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

Near an unstable equilibrium of a mechanical system, the linearised dynamics in suitable coordinates (x,y)(x,y) is generated by the vector field X=x ∂x−y ∂yX = x\,\partial_x - y\,\partial_y on R2\mathbb{R}^2. ((a)) Find the flow φt(x0,y0)\varphi_t(x_0,y_0) of XX. ((b)) A trajectory starts at (x0,y0)=(2,1)(x_0,y_0) = (2,1). Find its position at t=ln⁡2t = \ln 2.

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