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.
This sequence diagram outlines the mathematical derivation and physical verification steps detailed in the Derivation sheet, specifically focusing on the transition from identifying a non-conservative "vortex" field to establishing a conservative "radial" field and its potential function.
sequenceDiagram
autonumber
participant U as User / Learner
participant M as Mathematical Logic
participant P as Path Integration
participant C as Conservative Field Analysis
Note over U, P: Phase 1: Identifying Non-Conservative Fields
U->>M: Define Vortex Field: F = -ky i + kx j
M->>P: Calculate Work along Circular Path (Wa)
P-->>M: Wa = (Ï€/2) * k * r0^2
M->>P: Calculate Work along Straight Path (Wb)
P-->>M: Wb = k * r0^2
M->>M: Compare Wa ≠Wb
M->>M: Verify Curl: ∇ × F = 2k (≠0)
M-->>U: Result: Field is Non-Conservative
Note over U, C: Phase 2: Deriving the Conservative "Twin"
U->>M: Propose "Twin" Field: F = kx i + ky j
M->>M: Check Curl: ∇ × F = 0 (Conservative)
M->>C: Integrate ∂U/∂x = -kx
C-->>M: U(x,y) = -1/2 kx^2 + f(y)
M->>C: Differentiate U w.r.t y and match -ky
C-->>M: f'(y) = -ky -> f(y) = -1/2 ky^2
M-->>U: Final Potential Function: U = -1/2 k(x^2 + y^2)
Note over U, C: Phase 3: Verifying Path Independence
U->>P: Evaluate Work between A(r0,0) and B(0,r0)
P->>C: Check Potential Difference: ΔU = Ub - Ua
C-->>P: Ua = -1/2 kr0^2, Ub = -1/2 kr0^2
P-->>M: ΔU = 0 -> Work (W) = 0
M-->>U: Path Independence Confirmed (W=0 for any path)
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***Derivation Sheet***
Work Done by a Non-Conservative Force and Conservative Force@{ticket: 1st,assigned: Primary,priority: 'Very High'}
Simulating Conservative Fields and Path Independence@{assigned: SequenceDiagram}
***Resulmation***
The work Done Along a Circular Path and a Straight Line under non-conservative force and conservative force@{ticket: 2nd, assigned: Demostrate,priority: 'High'}
Work accumulation between Conservative and Non-Conservative@{assigned: Demostrate}
Gradient Ascent A Visual Study of Conservative Work@{assigned: Demostrate}
The work Done Along a Circular Path and a Straight Line under non-conservative force and conservative force@{assigned: Demo1}
Work accumulation between Conservative and Non-Conservative@{assigned: Demo2}
Gradient Ascent: A Visual Study of Conservative Work@{assigned: Demo3}
Visualizing Conservative Fields: From Path Dependence to Potential Surfaces@{assigned: StateDiagram}
***IllustraDemo***
Work Conservative and Non-Conservative Paths@{ticket: 3rd,priority: 'Low', assigned: Narrademo}
Path Matters Conservative vs Non-Conservative Forces@{assigned: Illustrademo}
Force Fields: Does the Path Matter@{assigned: Illustragram}
The Geometry of Force: Mapping Logic and Path Independence@{assigned: Seqillustrate}
***Ex-Demo***
The Mechanics of Path Dependency in Force Fields@{ticket: 4th, assigned: Flowscript,priority: 'Very High'}
Mechanics of Conservative and Non-Conservative Force Fields@{assigned: Flowchart}
Principles of Conservative and Non-Conservative Dynamics@{assigned: Mindmap}
***Narr-graphic***
Computational Dynamics of Force Fields@{ticket: 5th,assigned: Flowstra,priority: 'Very Low'}
The Topography of Force: Path Independence and Potential Fields@{assigned: Statestra}
Visual and Orchestra