{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Stability analysis\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/stability-analysis). 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": [ "Equilibrium means that the net force and net torque are zero. Stability asks a different\n", "question: what happens after a small disturbance? A stable equilibrium tends to remain\n", "nearby, while an unstable equilibrium develops motion away from the original state. A\n", "single perfectly balanced snapshot cannot answer this question.\n", "\n", "## Equilibrium comes first\n", "\n", "For one coordinate $x$ in a conservative\n", "system, force is related to potential energy by\n", "\n", "\n", "$$\n", "F(x)=-\\frac{dU}{dx}\n", "$$\n", "\n", "\n", "An equilibrium position\n", "$x_0$ satisfies\n", "\n", "\n", "$$\n", "F(x_0)=0\\qquad\\text{or equivalently}\\qquad U'(x_0)=0\n", "$$\n", "\n", "\n", "This condition alone does not classify the equilibrium. The bottom of a bowl and the top\n", "of a hill both have zero slope. Their responses to a displacement are opposite.\n", "\n", "For a rigid body, equilibrium also requires zero net torque. For a discretized elastic\n", "body, equilibrium means that internal forces, applied loads, and support reactions give a\n", "zero residual on every unconstrained degree of freedom.\n", "\n", "## The one-dimensional curvature test\n", "\n", "Near $x_0$, expand the potential\n", "energy in powers of the small displacement\n", "$\\eta=x-x_0$:\n", "\n", "\n", "$$\n", "U(x_0+\\eta)\\approx U(x_0)+\\frac12U''(x_0)\\eta^2\n", "$$\n", "\n", "\n", "The linear term vanishes because the point is an equilibrium. The nearby force is\n", "\n", "\n", "$$\n", "F(x_0+\\eta)\\approx-U''(x_0)\\eta\n", "$$\n", "\n", "\n", "If $U''(x_0)>0$, displacement\n", "raises the energy and the force points back toward equilibrium. This is a strict local\n", "minimum and is stable for the ideal conservative model. If\n", "$U''(x_0)<0$, one direction\n", "lowers energy and the force amplifies the displacement, so the point is unstable.\n", "\n", "If $U''(x_0)=0$, the test is\n", "inconclusive. Higher-order terms matter. For example, a positive fourth-power potential\n", "has a stable but unusually flat minimum, while a negative fourth-power potential is\n", "unstable." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import sympy as sp\n", "\n", "x = sp.Symbol(\"x\", real=True)\n", "energy = x**4 - 2 * x**2\n", "for equilibrium in sp.solve(sp.diff(energy, x), x):\n", " curvature = sp.diff(energy, x, 2).subs(x, equilibrium)\n", " print(equilibrium, \"stable\" if curvature > 0 else \"unstable\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The tolerance must reflect numerical error and units. Comparing a dimensional curvature\n", "directly with a unitless machine epsilon is not meaningful.\n", "\n", "## Small oscillations around a stable point\n", "\n", "For mass $m$, linearized motion near a stable\n", "one-dimensional equilibrium obeys\n", "\n", "\n", "$$\n", "m\\ddot{\\eta}+k_{\\mathrm{eff}}\\eta=0\\qquad k_{\\mathrm{eff}}=U''(x_0)\n", "$$\n", "\n", "\n", "The small-oscillation angular frequency is\n", "\n", "\n", "$$\n", "\\omega=\\sqrt{\\frac{k_{\\mathrm{eff}}}{m}}\n", "$$\n", "\n", "\n", "Thus stability and local oscillation frequency come from the same curvature. A smaller\n", "positive curvature gives slower oscillation and greater sensitivity to finite\n", "disturbances. The formula describes only motion sufficiently close to the equilibrium\n", "that nonlinear terms remain small.\n", "\n", "Damping changes what happens to the oscillation but does not turn a negative-stiffness\n", "equilibrium into an ordinary stable minimum. Positive damping causes motion near a stable\n", "minimum to decay. Negative damping injects energy and can destabilize an otherwise stable\n", "elastic mode.\n", "\n", "## Several degrees of freedom\n", "\n", "Let the small displacement vector be\n", "$\\boldsymbol\\eta$.\n", "Near equilibrium, the energy has the quadratic form\n", "\n", "\n", "$$\n", "U(\\mathbf q_0+\\boldsymbol\\eta)\\approx U(\\mathbf q_0)+\\frac12\\boldsymbol\\eta^{\\mathsf T}\\mathbf H\\boldsymbol\\eta\n", "$$\n", "\n", "\n", "The Hessian $\\mathbf H$ contains\n", "second derivatives of energy. For linear elasticity it plays the role of the tangent\n", "stiffness matrix. Stability is directional:\n", "\n", "- If every allowed non-rigid direction gives positive quadratic energy, the equilibrium\n", " is a strict local minimum.\n", "- If at least one allowed direction gives negative quadratic energy, the equilibrium is\n", " unstable in that direction.\n", "- A zero direction can be a harmless rigid motion, a mechanism, a symmetry, or a point\n", " where higher-order terms must decide the result.\n", "\n", "For dynamics with a positive mass matrix, normal modes satisfy\n", "\n", "\n", "$$\n", "\\mathbf H\\boldsymbol\\phi=\\omega^2\\mathbf M\\boldsymbol\\phi\n", "$$\n", "\n", "\n", "A negative value of\n", "$\\omega^2$ means the\n", "mode grows exponentially instead of oscillating. A positive value produces an\n", "oscillatory mode in the undamped linear model. Constraints must be applied consistently\n", "before interpreting zero or negative eigenvalues." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "\n", "stiffness = np.array([[2.0, -1.0], [-1.0, 2.0]])\n", "masses = np.array([1.0, 2.0])\n", "inverse_sqrt_mass = np.diag(1 / np.sqrt(masses))\n", "weighted_stiffness = inverse_sqrt_mass @ stiffness @ inverse_sqrt_mass\n", "frequency_squared, modes = np.linalg.eigh(weighted_stiffness)\n", "print(\"Squared angular frequencies (1/s^2):\", frequency_squared)\n", "assert np.all(frequency_squared > 0)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The eigenvector or mode shape matters as much as the eigenvalue because it identifies the\n", "physical deformation that loses stability.\n", "\n", "## Buckling as loss of stability\n", "\n", "A slender compressed member may satisfy static force balance while its straight shape\n", "becomes unstable to sideways bending. For an ideal pinned-pinned column, Euler's critical\n", "load is\n", "\n", "\n", "$$\n", "P_{\\mathrm{cr}}=\\frac{\\pi^2EI}{L^2}\n", "$$\n", "\n", "\n", "Here $E$ is Young's modulus,\n", "$I$ describes cross-sectional\n", "bending resistance, and $L$ is the\n", "unsupported length for this end condition. Other end conditions replace\n", "$L$ by an effective length.\n", "\n", "The formula assumes an initially straight, slender, elastic column, small lateral\n", "deflection at the onset, a centered compressive load, and ideal supports. Real\n", "imperfections usually cause lateral deflection before the ideal critical load, so design\n", "must not treat the formula as an exact failure guarantee.\n", "\n", "In a numerical model, buckling appears when the tangent stiffness develops a zero mode\n", "and then a negative direction. Near this point, ordinary load-controlled Newton\n", "iterations can struggle because the tangent matrix is nearly singular. Small geometric\n", "imperfections, displacement control, or continuation methods can help trace the physical\n", "post-buckling path.\n", "\n", "## Stable does not always mean safe\n", "\n", "Stability is local unless a larger region has been analyzed. A marble can rest in a\n", "shallow local minimum yet escape over a nearby barrier after a finite push. The energy\n", "barrier, not merely local curvature, controls this finite-disturbance behavior.\n", "\n", "Metastable states are local minima that are not the global minimum. Snap-through systems\n", "can jump from one stable branch to another when a limit point is crossed. Structures may\n", "also remain stable but exceed allowable stress, strain, displacement, fatigue, or damage\n", "limits. Stability analysis and strength analysis answer different questions.\n", "\n", "For nonconservative systems, an energy minimum may not be available. Follower forces,\n", "active control, frictional effects, or time-dependent loading can require direct\n", "linearization of the equations of motion. In a first-order system\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=\\mathbf g(\\mathbf y)\n", "$$\n", "\n", "\n", "the local perturbation satisfies\n", "\n", "\n", "$$\n", "\\dot{\\boldsymbol\\eta}=\\mathbf J\\boldsymbol\\eta\n", "$$\n", "\n", "\n", "where $\\mathbf J$ is evaluated at\n", "the equilibrium. Eigenvalues with positive real parts give growing perturbations.\n", "Eigenvalues with negative real parts give local decay. Purely imaginary or zero real\n", "parts make the linear test incomplete and demand additional analysis.\n", "\n", "## Numerical perturbation tests\n", "\n", "Analytic curvature and eigenvalue tests should be paired with direct checks. Apply a\n", "small displacement in a chosen direction and evaluate the force without advancing the\n", "state. For a stable elastic mode, the force component along that displacement should be\n", "restoring. Repeat with half the disturbance. In the linear regime, force should halve\n", "and the inferred stiffness should approach a constant." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "for amplitude in [0.1, 0.01, 0.001]:\n", " for direction in np.eye(2):\n", " displacement = amplitude * direction\n", " force = -stiffness @ displacement\n", " projected_force = np.dot(force, direction)\n", " assert projected_force < 0\n", " print(amplitude, projected_force)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Use both positive and negative perturbations. An asymmetric response may reveal\n", "nonlinearity, unilateral contact, or an equilibrium that was not solved accurately. Do\n", "not use perturbations so small that roundoff dominates or so large that they leave the\n", "local regime.\n", "\n", "## Separating physical and numerical instability\n", "\n", "Physical instability remains when the spatial and temporal discretization are refined.\n", "Numerical instability changes or disappears when the time step, solver, tolerance, or\n", "mesh changes. To distinguish them, first solve the equilibrium residual accurately,\n", "inspect the tangent eigenvalues, and then rerun dynamic perturbations at smaller time\n", "steps with an appropriate integration method.\n", "\n", "Energy provides another distinction. Growth along a genuine negative-curvature direction\n", "is consistent with decreasing potential energy. Unexplained growth in total energy for a\n", "closed conservative model points instead to integration or force-evaluation error.\n", "\n", "A reliable stability claim therefore names the equilibrium, allowed perturbations,\n", "constraints, parameter range, and mathematical test. It also states whether the claim is\n", "local or includes finite disturbances. That description is more informative than a\n", "single label such as “stable.”\n", "\n", "## A worked Python notebook\n", "\n", "Compare local curvatures of the potential x to the fourth power minus twice x squared. The question for this calculation is precise: how does **energy curvature**\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 negative curvature at a stationary point indicates a direction of instability. 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", "-1.3 to 1.3 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", "energy curvature, expressed in N/m. 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", "$$\n", "f(q) = 12 q^{2} - 4\n", "$$\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 1 m, the expression gives 8 N/m. 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-009", "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(\"12*q**2-4\", locals={\"q\": q})\n", "reference_input = 1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted energy curvature:\", reference_output, \"N/m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "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", "16.28 N/m. The midpoint, at 0 m, gives\n", "-4 N/m. The upper endpoint gives 16.28 N/m.\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-011", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-1.3, 0.0, 1.3])\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} N/m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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 energy curvature. 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-013", "metadata": {}, "source": [ "input_grid = np.linspace(-1.3, 1.3, 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(\"Energy curvature (N/m)\")\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-014", "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", "energy curvature change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1 m is\n", "24 in those ratio units. Locally, the output increases 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", "$$\n", "f'(q) = 24 q\n", "$$\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-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.013\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, \"N/m\")\n", "print(\"Full expression:\", full_prediction, \"N/m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"N/m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 1.013 m. The tangent estimate is\n", "8.312 N/m, while the full expression gives\n", "8.31403 N/m. 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-017", "metadata": {}, "source": [ "input_span = 2.6\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-018", "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 energy curvature 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 negative curvature at a stationary point indicates a direction of instability. 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 N/m:\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-019", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.3) - evaluate(-1.3))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-1.3, 1.3, 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-020", "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 15.1, \"Simple Harmonic Motion\"](https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 13, \"Energy, Kinetic Energy, and Work\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter13/).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 26, \"Elastic Properties of Materials\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter26/).\n", "- [Benjamin Crowell, *Light and Matter*, equilibrium and small-oscillation discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Elasticity\"](https://physics.info/elasticity/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "stability-analysis.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/stability-analysis", "source_sha256": "1b84d976e758cccfd75cbb0efa358326148f8734fca8eaabe0ca214b539dfaf3" } }, "nbformat": 4, "nbformat_minor": 5 }