{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Potential energy landscapes\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/potential-energy-landscapes). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A potential-energy landscape assigns energy to each configuration of a conservative\n", "system. Its height is energy, not physical altitude unless the model specifically uses\n", "gravity. Slopes determine force, intersections with total energy locate turning points,\n", "and local curvature helps classify equilibrium.\n", "\n", "## Potential and force\n", "\n", "In one coordinate,\n", "\n", "\n", "$$\n", "F(x)=-\\frac{dU}{dx}\n", "$$\n", "\n", "\n", "A potential that rises toward positive\n", "$x$ has a force toward negative\n", "$x$. A flat potential gives zero force,\n", "even if its absolute energy value is large.\n", "\n", "Adding a constant changes no force:\n", "\n", "\n", "$$\n", "-\\frac{d}{dx}[U(x)+C]=-\\frac{dU}{dx}\n", "$$\n", "\n", "\n", "Therefore the vertical zero on a potential plot is conventional. Energy differences,\n", "slopes, and allowed motion are physical within the model; the chosen additive constant\n", "must be stated when values are compared.\n", "\n", "## Energy determines allowed positions\n", "\n", "For one particle of mass\n", "$m$ with total mechanical energy\n", "$E$,\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+U(x)\n", "$$\n", "\n", "\n", "so\n", "\n", "\n", "$$\n", "v^2=\\frac{2}{m}[E-U(x)]\n", "$$\n", "\n", "\n", "Classical motion is allowed only where\n", "$E\\ge U(x)$. Where\n", "$E0\\;\\text{stable minimum},\\qquad U''(x_*)<0\\;\\text{unstable maximum}\n", "$$\n", "\n", "\n", "A local minimum resists a small displacement because force points back toward it. A\n", "local maximum pushes a displaced state farther away. If\n", "$U''(x_*)=0$, higher-order terms\n", "or a direct force analysis are required.\n", "\n", "“Stable” here is local and refers to the ideal conservative model. Enough energy can\n", "carry a state over a finite barrier and out of a local well.\n", "\n", "## Harmonic approximation near a minimum\n", "\n", "Expand a smooth potential near a stable equilibrium\n", "$x_*$:\n", "\n", "\n", "$$\n", "U(x)\\approx U(x_*)+\\frac12U''(x_*)(x-x_*)^2\n", "$$\n", "\n", "\n", "The linear term vanishes at equilibrium. Comparing with a spring potential gives\n", "effective stiffness\n", "$k_{\\mathrm{eff}}=U''(x_*)$\n", "and small-oscillation angular frequency\n", "\n", "\n", "$$\n", "\\omega=\\sqrt{\\frac{U''(x_*)}{m}}\n", "$$\n", "\n", "\n", "This approximation applies only while higher-order terms remain small. Oscillation\n", "frequency can depend on amplitude outside that region.\n", "\n", "## Example: a symmetric double well\n", "\n", "Consider\n", "\n", "\n", "$$\n", "U(x)=a(x^2-b^2)^2,\\qquad a>0,\\quad b>0\n", "$$\n", "\n", "\n", "Its derivative is\n", "\n", "\n", "$$\n", "U'(x)=4ax(x^2-b^2)\n", "$$\n", "\n", "\n", "so stationary points occur at\n", "$x=0$ and\n", "$x=\\pm b$. The two outer\n", "points are minima with zero potential under this convention. The center is a maximum\n", "with barrier energy\n", "$U(0)=ab^4$.\n", "\n", "For $Eab^4$, the allowed set\n", "connects across the center. At exactly the barrier energy, a state placed at the unstable\n", "top with zero velocity remains there in the ideal exact model, but any perturbation or\n", "roundoff can send it toward one well." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import sympy as sp\n", "\n", "x = sp.Symbol(\"x\", real=True)\n", "potential = (x**2 - 1) ** 2 # joules when x is expressed in metres\n", "energy = sp.Rational(1, 4)\n", "roots = sorted(float(r) for r in sp.solve(sp.Eq(potential, energy), x))\n", "allowed = []\n", "for left, right in zip(roots[:-1], roots[1:]):\n", " if float(potential.subs(x, (left + right) / 2)) <= float(energy):\n", " allowed.append((left, right))\n", "print(\"Turning points (m):\", roots)\n", "print(\"Allowed intervals (m):\", allowed)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Root finding alone is insufficient: candidate roots need domain checks, and intervals\n", "between them must be classified.\n", "\n", "## Barriers, wells, and escape\n", "\n", "A well is a region around a local minimum. A barrier is a local high region separating\n", "allowed regions at lower energy. Escape requires enough total energy to reach a path out\n", "of the well in the conservative model.\n", "\n", "In more than one dimension, the lowest escape route may pass through a saddle rather\n", "than the highest-looking point on a chosen one-dimensional slice. A slice can conceal\n", "directions in configuration space and should not be treated as the full landscape.\n", "\n", "## Multiple dimensions\n", "\n", "For coordinates\n", "$\\mathbf q=(q_1,\\ldots,q_n)$,\n", "\n", "\n", "$$\n", "\\mathbf F=-\\nabla U(\\mathbf q)\n", "$$\n", "\n", "\n", "Equilibria satisfy\n", "$\\nabla U=\\mathbf 0$. The\n", "Hessian matrix contains second derivatives:\n", "\n", "\n", "$$\n", "H_{ij}=\\frac{\\partial^2U}{\\partial q_i\\partial q_j}\n", "$$\n", "\n", "\n", "At an unconstrained equilibrium with ordinary positive-definite kinetic energy, a\n", "positive-definite Hessian indicates a strict local minimum. A negative eigenvalue gives\n", "an unstable direction. Mixed positive and negative eigenvalues describe a saddle.\n", "\n", "For constrained systems, stability must be tested only along allowed tangent directions.\n", "A potential may slope in a forbidden normal direction without driving allowed motion.\n", "\n", "## Sampling a potential numerically\n", "\n", "A tabulated potential needs interpolation. Interpolating values and separately\n", "interpolating force can make\n", "$\\mathbf F=-\\nabla U$ fail.\n", "Where possible, define one smooth interpolant for potential and differentiate it, or use\n", "a construction designed to preserve the relationship." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "\n", "positions = np.linspace(-1.5, 1.5, 301)\n", "values = (positions**2 - 1) ** 2\n", "force_estimate = -np.gradient(values, positions, edge_order=2)\n", "force_exact = -4 * positions * (positions**2 - 1)\n", "print(\n", " \"Largest interior force error (N):\",\n", " np.max(abs(force_estimate[1:-1] - force_exact[1:-1])),\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Grid spacing must resolve narrow wells and barriers. A coarse grid can miss a stationary\n", "point or merge distinct roots. Refine and check whether extrema and barrier heights\n", "stabilize.\n", "\n", "## Root-finding care\n", "\n", "Turning-point equations can have repeated roots, closely spaced roots, and roots at\n", "domain boundaries. Sign-change bracketing misses an even-multiplicity root that touches\n", "zero without crossing it. Derivative information, interval methods, or symbolic\n", "candidates may be needed.\n", "\n", "Every root must be substituted back into the original potential equation. Squaring,\n", "cancelling, or changing variables can introduce invalid candidates or discard domain\n", "restrictions.\n", "\n", "## Time-dependent and dissipative systems\n", "\n", "A time-dependent potential\n", "$U(\\mathbf q,t)$ is a changing\n", "landscape. Even when force is its negative spatial gradient,\n", "\n", "\n", "$$\n", "\\frac{d}{dt}(K+U)=\\frac{\\partial U}{\\partial t}\n", "$$\n", "\n", "\n", "so a fixed horizontal total-energy line no longer describes all future allowed positions.\n", "\n", "With friction or drag, mechanical energy changes and a trajectory moves between energy\n", "contours. The potential remains useful for conservative forces, but it is not a complete\n", "dynamics description.\n", "\n", "## Visualization rules\n", "\n", "A useful landscape plot includes coordinate units, energy units, potential zero,\n", "parameter values, and the total-energy level being considered. Mark stationary points\n", "and turning points numerically. Do not allow automatic vertical scaling to imply that\n", "two landscapes have comparable barrier sizes when their axes differ.\n", "\n", "For two-dimensional configuration spaces, contour plots show equal-potential curves.\n", "Force is perpendicular to contours and points toward lower potential, but unequal plot\n", "axis scales can distort visual perpendicularity.\n", "\n", "## Validation checks\n", "\n", "- Adding a constant to potential leaves force and motion unchanged.\n", "- Numerical force agrees with the negative potential gradient.\n", "- Allowed regions satisfy\n", " $E\\ge U$.\n", "- Turning points satisfy the original equation and physical domain.\n", "- Local minima, maxima, and saddles agree with derivative or Hessian tests.\n", "- The harmonic approximation is compared with the exact potential over a stated range.\n", "- Grid refinement preserves important extrema, roots, and barrier heights.\n", "- Potential and energy use the same zero convention.\n", "- Constraint directions and time dependence are included where relevant.\n", "- A landscape is not mistaken for a literal spatial track.\n", "\n", "A potential-energy landscape compresses conservative dynamics into geometry in\n", "configuration space. Reading it correctly requires slopes, domains, energy levels, and\n", "system assumptions—not only the shape of a plotted curve.\n", "\n", "## A worked Python notebook\n", "\n", "Explore two potential wells separated by a central barrier. The question for this calculation is precise: how does **double-well potential**\n", "change when we change **position**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "A particle can visit only positions whose potential does not exceed its total energy. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "-1.5 to 1.5 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for position, expressed in m. The output is\n", "double-well potential, expressed in J. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\left(q^{2} - 1\\right)^{2}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.5 m, the expression gives 0.5625 J. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"(q**2-1)**2\", locals={\"q\": q})\n", "reference_input = 0.5\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted double-well potential:\", reference_output, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "1.5625 J. The midpoint, at 0 m, gives\n", "1 J. The upper endpoint gives 1.5625 J.\n", "Read these together with the assumptions: they describe different choices of position\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-1.5, 0.0, 1.5])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} J\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows position; the vertical axis shows double-well potential. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(-1.5, 1.5, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Position (m)\")\n", "ax.set_ylabel(\"Double-well potential (J)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "double-well potential change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.5 m is\n", "-1.5 in those ratio units. Locally, the output decreases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 4 q \\left(q^{2} - 1\\right)\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.515\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"J\")\n", "print(\"Full expression:\", full_prediction, \"J\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 0.515 m. The tangent estimate is\n", "0.54 J, while the full expression gives\n", "0.539894 J. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 3.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "position inside the stated interval, predict whether double-well potential will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: A particle can visit only positions whose potential does not exceed its total energy. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of J:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.5) - evaluate(-1.5))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-1.5, 1.5, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 8.1, \"Potential Energy of a System\"](https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system).\n", "- [OpenStax, *University Physics Volume 1*, Section 8.4, \"Potential Energy Diagrams and Stability\"](https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 14, Sections 14.4–14.6](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter14/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 14, \"Conservation of Energy\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Potential Energy\"](https://physics.info/energy-potential/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "potential-energy-landscapes.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/potential-energy-landscapes", "source_sha256": "7c84bf6254946e0b034205bb8e1fd1ef6f6a0ee19d97ccaff3d7d6cef8032913" } }, "nbformat": 4, "nbformat_minor": 5 }