請問什麼是power law potentials? 我知道potential energy, U i.e., gravitational potential U=U(x)=mgx above the earth ...eq(1) spring-mass potential U=U(x)=0.5kx^2 ... eq(2) it's well-known that (From wiki) Given an arbitrary potential energy function V(x), one can do a Taylor expansion in terms of x around an energy minimum (x = x0) to model the behavior of small perturbations from equilibrium. V(x)=V(x0)+V'(x0)*(x-x0)+0.5V''(x0)*(x-x0)^2+o(3) ... eq(3) 但是這個power law potentials, U(x-x0)=a*(x-x0)^b,以x0為零點 是如何證明的? 跟Taylor有關係嗎? 由eq(1),eq(2)知道跟距離的冪次有關, eq(2)可以由eq(3)得,如下 Because V(x0) is a minimum, the first derivative evaluated at x0 must be zero, so the linear term drops out, V'(0)=0 The constant term V(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved: V(x)~0.5V''(0)*(x)^2=0.5kx^2 想請教 如何從此得知eq(1)? 相關的物理意義是甚麼意思? 不才懇請諸賢指點 謝謝 叩首萬謝 ToT -- 網誌 http://www.wretch.cc/blog/harry901 (Mathematical puzzles, Physics, econophysics, education, and some travel notes.) --

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