M043 · Work and energy

Conservative Forces & Path Dependence

Edit two paths between shared endpoints. Separate potential work, rotational circulation and sliding loss; test acquired field-force samples against the geometric loop.

Analytic field classification $\text{Ready}$

First route field work $\text{Ready}$

Second route field work $\text{Ready}$

First route with resistance $\text{Ready}$

Second route with resistance $\text{Ready}$

First route length $\text{Ready}$

Second route length $\text{Ready}$

Closed-loop field work $\text{Ready}$

Closed-loop resistance work $\text{Ready}$

Closed-loop total work $\text{Ready}$

Oriented loop area $\text{Ready}$

Curl times oriented area $\text{Ready}$

Length change at double panels $\text{Ready}$

Selected route potential-part work $\text{Ready}$

Selected tangent $\text{Ready}$

Acquired force stations $\text{Ready}$

Acquired closed-loop field work $\text{Ready}$

Acquired minus reference loop $\text{Ready}$

PATH / FIELD / LOOP

Keep the endpoints; change the journey

Radial spring coefficient

k

0.70\,\frac{\mathrm N}{\mathrm m}

Uniform downward force

G

1.00\,N

Rotational field coefficient

\gamma

0.00\,\frac{\mathrm N}{\mathrm m}

Sliding resistance magnitude

f

0.00\,N

Route to edit; first or second

j

0.00\,

Reverse both routes

\sigma

0.00\,

Length quadrature panels

n_q

8.00\,

First path control 1 horizontal

P^{()}_{1,x}

-2.00\,m

First path control 2 horizontal

P^{()}_{2,x}

2.00\,m

First path control 1 height

P^{()}_{1,y}

3.00\,m

First path control 2 height

P^{()}_{2,y}

3.00\,m

Second path control 1 horizontal

P^{()}_{1,x}

-2.00\,m

Second path control 2 horizontal

P^{()}_{2,x}

2.00\,m

Second path control 1 height

P^{()}_{1,y}

-3.00\,m

Second path control 2 height

P^{()}_{2,y}

-3.00\,m

Replay fraction

\tau

0.45\,

Force scan intervals

n_s

24.00\,

Force noise bound

\eta_F

0.00\,N

Two planar cubic Bézier paths share fixed endpoints at minus two, minus one and two, one metres. Each has two freely editable inner points. Coordinates are right-handed: horizontal right, vertical up, depth out of the path plane. The 3D paths and all integrals use these same physical coordinates.

The spring and uniform downward parts have a known potential. The rotational part has constant nonzero curl when its coefficient is nonzero. A zero result on one loop or two matching paths does not prove a global potential. The classification comes from the declared analytic field.

The paths are prescribed by an external drive, not solved trajectories under the plotted forces. No kinetic energy is inferred from the path parameter. Sliding resistance has a specified constant magnitude opposing the local traversal direction; static friction, contact normal loads and drive dynamics are omitted. Its direction is undefined at a stationary tangent, and that arrow is omitted.

The directed loop follows the first path forward and the second backward. Reversing a position-dependent field changes its work sign; resistance remains negative. Curl times oriented area applies only to the field contribution. Self-intersections use signed winding area.

Composite eight-point Gauss quadrature integrates the smooth field work and path length. Field-work integrands are polynomial; length is numerical, especially near stationary tangents. The change at double panels is a diagnostic, not an error bound. Plot prefixes use a separate dense partition.

Raw acquisition records exact positions and noisy field-force components on both paths. Trapezoidal force–displacement integration uses these records only. Resistance work is a separate model calculation and is not claimed as measured. Replay, route selection, resistance and quadrature changes preserve the field record; geometry, direction, field or sensor changes clear it.

Prescribed paths; field work and sliding dissipation are separate. No drive or contact dynamics.

Start with the essentials

Focus question: Does the same destination require the same work?
One-sentence intuition: One zero loop does not establish a global potential.

Core mathematical model

Known conservative and rotational parts

\begin{aligned}\mathbf F&=(-kx-\gamma y,\,-G-ky+\gamma x)\\U_c&=\tfrac12k(x^2+y^2)+Gy\end{aligned}

Only the spring and uniform downward parts are generated by this potential.

A path integral projects onto the tangent

W_C=\int_0^1\mathbf F(\mathbf c(q))\cdot\mathbf c^{\prime}(q)\,\mathrm dq

The path parameter is not time. The apparatus is driven along the selected path.

Oriented loop and curl

\begin{aligned}A_{\mathrm{or}}&=\tfrac12\oint(x\,\mathrm dy-y\,\mathrm dx)\\\oint\mathbf F\cdot\mathrm d\mathbf r&=2\gamma A_{\mathrm{or}}\end{aligned}

The first path forward and second backward define orientation. Self-intersections contribute with winding sign.

Dissipation does not reverse sign

\begin{aligned}W_f&=-fL\\W_{f,\mathrm{loop}}&=-f(L_A+L_B)\end{aligned}

Resistance depends on traversal direction, so its loop work is not a curl integral of one position-only field.

Raw force–displacement estimate

\widehat W_C=\sum_i\frac{\mathbf F_i^{\mathrm{obs}}+\mathbf F_{i+1}^{\mathrm{obs}}}{2}\cdot(\mathbf r_{i+1}-\mathbf r_i)

Positions are exact in this simulated acquisition; field-force components carry bounded deterministic noise.

Common difficulties

Model and evidence

Typical misconceptionReversing a loop always reverses every work contribution.

Better mental modelPosition-only field work reverses; sliding resistance follows the new tangent and still dissipates.

Run the experiment

01
Keep endpoints fixed
Edit both internal points on each route.
What to observe: Potential-part work stays fixed, even when path length changes.
02
Give the field circulation
Choose Rotational field and change its sign.
What to observe: Closed-loop field work follows oriented area and reverses with the loop.
03
Add resistance
Set sliding resistance, then reverse both routes.
What to observe: The field term changes sign while the resistance term remains negative.
04
Acquire a geometric integral
Acquire field forces; increase scan intervals and compare with the model.
What to observe: Measured loop work is a force–displacement sum; resistance is still a separate calculation.