Consider a particle of mass m moving in one dimensional potential V(x)=x4 – 2x2. The phase trajectory (x,p) of the particle is best represented by
Phase trajectories are the constant-energy curves \(\frac{p^2}{2m}+V(x)=E\), so their shape is fixed by the extrema of V(x).
\(V'(x)=4x^3-4x=4x(x^2-1)=0\) gives x = 0 and x = ±1. Since V(0) = 0 is a local maximum and V(±1) = –1 are minima, this is a symmetric double-well potential.
The minima (±1, 0) are stable equilibria (centres): for –1 < E < 0 the particle is trapped in one well, giving two separate families of closed orbits around x = +1 and x = –1.
The maximum (0, 0) is an unstable equilibrium (saddle point). The E = 0 curve, p = ±√(2m(2x² – x⁴)), is the separatrix – a figure-of-eight passing through the origin and enclosing both wells.
For E > 0 the trajectories are large curves enclosing both wells outside the separatrix. The portrait with two centres, a saddle at the origin and a separatrix is the one that matches; portraits made of concentric closed curves around a single centre describe a single-well (harmonic-type) potential, and straight lines crossing at the origin describe only an inverted-oscillator saddle.
Hence, the answer is Phase portrait (B) — refer to the image in the original question.
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
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(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
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List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
