The central conclusion of the analysis is that the force field, $F=k\left(x^1 e_2-x^2 e_1\right)$, is not conservative. This was established through two pieces of evidence: first, the work done on the particle was path-dependent, yielding $W_a=\frac{\pi}{2} k r_0^2$ for the circular path and $W_b=k r_0^2$ for the straight line path. Second, the non-conservative nature was confirmed by calculating the curl of the force field, $\nabla \times F$, which resulted in a non-zero constant vector, $2 k e_3$. This nonzero curl value demonstrates the presence of constant circulation or vorticity in the field, which is characteristic of a rotational (non-conservative) force.
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title The Path-Field Paradox
Resulmation: The work Done Along a Circular Path and a Straight Line under non-conservative force and conservative force
: Work accumulation between Conservative and Non-Conservative
: Gradient Ascent A Visual Study of Conservative Work
IllustraDemo: Work Conservative and Non-Conservative Paths
Ex-Demo: The Mechanics of Path Dependency in Force Fields
Narr-graphic: Computational Dynamics of Force Fields
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