{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Stability and equilibrium points\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/stability-equilibrium-points). 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": [ "An equilibrium is a state that can remain unchanged under the model. Stability asks what\n", "happens after a small disturbance. The two ideas are related but not identical: a ball\n", "at the top of a hill is in equilibrium in the ideal model, yet an arbitrarily small\n", "displacement makes it move away.\n", "\n", "## Equilibrium in one dimension\n", "\n", "For a particle governed by\n", "\n", "\n", "$$\n", "m\\ddot x=F(x)\n", "$$\n", "\n", "\n", "an equilibrium position\n", "$x_*$ satisfies\n", "\n", "\n", "$$\n", "F(x_*)=0\n", "$$\n", "\n", "\n", "The complete stationary state also has\n", "$v=0$. Passing through a zero-force point\n", "with nonzero velocity is not equilibrium; inertia carries the particle onward.\n", "\n", "If force derives from potential energy,\n", "$F=-dU/dx$,\n", "then equilibria occur at stationary points:\n", "\n", "\n", "$$\n", "U'(x_*)=0\n", "$$\n", "\n", "\n", "Root candidates must lie in the physical domain and satisfy the original equation.\n", "\n", "## Stable, unstable, and neutral behavior\n", "\n", "At a smooth isolated stationary point, potential curvature gives the local classification:\n", "\n", "\n", "$$\n", "U''(x_*)>0\\Rightarrow\\text{stable},\\qquad U''(x_*)<0\\Rightarrow\\text{unstable}\n", "$$\n", "\n", "\n", "A strict local minimum raises potential for nearby displacement, producing a restoring\n", "force. A local maximum lowers potential on either side, so force pushes away. If\n", "$U''(x_*)=0$,\n", "the test is inconclusive.\n", "\n", "For a perfectly flat potential over an interval, any stationary point is neutrally\n", "stable in position: a small displacement can remain at the new point rather than return.\n", "With nonzero perturbed velocity, the particle drifts because no restoring force acts.\n", "Stability must specify which state variables are perturbed.\n", "\n", "## Linearization near equilibrium\n", "\n", "Let $x=x_*+\\eta$\n", "with small $\\eta$. Expanding\n", "force gives\n", "\n", "\n", "$$\n", "F(x_*+\\eta)\\approx F(x_*)+F'(x_*)\\eta=F'(x_*)\\eta\n", "$$\n", "\n", "\n", "Since $F'=-U''$,\n", "\n", "\n", "$$\n", "m\\ddot\\eta+U''(x_*)\\eta=0\n", "$$\n", "\n", "\n", "For positive curvature, this is a harmonic oscillator with\n", "\n", "\n", "$$\n", "\\omega=\\sqrt{\\frac{U''(x_*)}{m}}\n", "$$\n", "\n", "\n", "For negative curvature, solutions contain exponential growth rather than oscillation.\n", "The linear approximation is local; higher-order terms matter after perturbations become\n", "large." ] }, { "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\n", "for root in sp.solve(sp.diff(potential, x), x):\n", " curvature = sp.diff(potential, x, 2).subs(x, root)\n", " kind = (\n", " \"stable\" if curvature > 0 else \"unstable\" if curvature < 0 else \"inconclusive\"\n", " )\n", " print(root, curvature, kind)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Numerical derivative tolerances need units: slope has force dimensions and curvature has\n", "force per length.\n", "\n", "## When the second derivative vanishes\n", "\n", "Consider $U(x)=ax^4$\n", "with $a>0$. At the origin, both first and\n", "second derivatives vanish, yet the potential has a stable minimum. Higher-order behavior\n", "classifies it.\n", "\n", "For $U(x)=ax^3$, first and\n", "second derivatives vanish at the origin, but the potential decreases on one side and\n", "increases on the other; it is not a stable minimum. Treating every zero curvature as\n", "neutral would misclassify both examples.\n", "\n", "## Damping changes asymptotic behavior\n", "\n", "With linear damping near a stable minimum,\n", "\n", "\n", "$$\n", "m\\ddot\\eta+b\\dot\\eta+k_{\\mathrm{eff}}\\eta=0\n", "$$\n", "\n", "\n", "where\n", "$k_{\\mathrm{eff}}=U''(x_*)$.\n", "Positive damping causes perturbations to decay toward equilibrium. Without damping, an\n", "ideal conservative system near a minimum usually oscillates rather than asymptotically\n", "settling.\n", "\n", "Negative damping adds energy and can destabilize an otherwise restoring potential.\n", "Potential shape alone therefore cannot classify systems with velocity-dependent or\n", "active forces.\n", "\n", "## General dynamical-system stability\n", "\n", "Write a first-order autonomous system as\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=\\mathbf f(\\mathbf y)\n", "$$\n", "\n", "\n", "An equilibrium state satisfies\n", "\n", "\n", "$$\n", "\\mathbf f(\\mathbf y_*)=\\mathbf 0\n", "$$\n", "\n", "\n", "For a small perturbation\n", "$\\boldsymbol\\eta=\\mathbf y-\\mathbf y_*$,\n", "linearization gives\n", "\n", "\n", "$$\n", "\\dot{\\boldsymbol\\eta}=\\mathbf J(\\mathbf y_*)\\boldsymbol\\eta\n", "$$\n", "\n", "\n", "where\n", "$J_{ij}=\\partial f_i/\\partial y_j$.\n", "Eigenvalues of the Jacobian determine local linear behavior. Any eigenvalue with positive\n", "real part gives an unstable direction. All eigenvalues with negative real parts give an\n", "asymptotically stable equilibrium. Purely imaginary or zero real parts make linearization\n", "inconclusive about some nonlinear cases.\n", "\n", "## Example: damped oscillator state matrix\n", "\n", "For\n", "$m\\ddot x+b\\dot x+kx=0$,\n", "choose state $\\mathbf y=(x,v)$.\n", "Then\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=\\begin{bmatrix}0&1\\\\-k/m&-b/m\\end{bmatrix}\\mathbf y\n", "$$\n", "\n", "\n", "The characteristic equation is\n", "\n", "\n", "$$\n", "\\lambda^2+\\frac bm\\lambda+\\frac km=0\n", "$$\n", "\n", "\n", "For positive mass, stiffness, and damping, both eigenvalues have negative real parts, so\n", "the origin is asymptotically stable. With zero damping they are purely imaginary and the\n", "ideal motion is bounded but does not decay.\n", "\n", "## Multiple-dimensional potential\n", "\n", "For generalized coordinates\n", "$\\mathbf q$, conservative\n", "equilibria satisfy\n", "$\\nabla U(\\mathbf q_*)=\\mathbf0$.\n", "The Hessian is\n", "\n", "\n", "$$\n", "H_{ij}=\\left.\\frac{\\partial^2U}{\\partial q_i\\partial q_j}\\right|_{\\mathbf q_*}\n", "$$\n", "\n", "\n", "A positive-definite Hessian indicates a strict local minimum for unconstrained\n", "coordinates. A negative eigenvalue gives a downhill unstable direction. Mixed signs\n", "indicate a saddle. Zero eigenvalues require symmetry, constraint, or higher-order\n", "analysis." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "\n", "x, y = sp.symbols(\"x y\", real=True)\n", "potential_2d = x**2 - y**2\n", "hessian = sp.hessian(potential_2d, (x, y))\n", "eigenvalues = np.linalg.eigvalsh(np.array(hessian, dtype=float))\n", "print(\"Curvatures:\", eigenvalues)\n", "print(\"A negative curvature identifies an unstable direction.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The Hessian should be symmetric for a sufficiently smooth scalar potential; substantial\n", "asymmetry in a numerical estimate is a diagnostic.\n", "\n", "## Constraints restrict perturbations\n", "\n", "A bead on a wire may be stable along the wire even if the ambient potential decreases\n", "in a direction the bead cannot move. Stability must be tested on the constraint\n", "manifold, using tangent perturbations that satisfy the linearized constraint.\n", "\n", "Reaction forces can also change the linearized dynamics. Simply checking the full-space\n", "Hessian without projecting into allowed coordinates may invent an unstable forbidden\n", "direction or miss a constraint-induced mode.\n", "\n", "## Driven and time-dependent systems\n", "\n", "A periodically driven system may have no time-independent equilibrium. It can instead\n", "have a periodic steady response whose stability is tested by perturbing the whole\n", "time-dependent solution. Floquet multipliers or a stroboscopic map can classify such\n", "orbits. Calling one instantaneous zero-acceleration point an equilibrium ignores the\n", "drive phase and velocity state.\n", "\n", "## Numerical equilibrium finding\n", "\n", "Root solvers can find\n", "$F(x)=0$ or\n", "$\\nabla U=0$. Their success\n", "flag is not classification. Verify the residual, domain, and nearby behavior.\n", "\n", "Newton's root method can converge to different roots depending on the initial guess and\n", "can fail where derivatives are small. Bracketing is safer in one dimension for sign-\n", "changing roots, but it can miss a force that only touches zero. Scan, bracket, refine,\n", "deduplicate, and compare with analytic structure where possible.\n", "\n", "## Perturbation experiments" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "for perturbation in [0.1, 0.01, 0.001]:\n", " for sign in [-1, 1]:\n", " position = 1.0 + sign * perturbation\n", " force = -4 * position * (position**2 - 1)\n", " print(f\"Offset {sign*perturbation:+.3f} m: force {force:+.6f} N\")\n", " assert force * (position - 1.0) < 0\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "A finite-time bounded trace is evidence, not proof of global stability. Numerical\n", "damping can make an unstable or neutral system look attracting, while numerical\n", "instability can make a stable system appear to diverge.\n", "\n", "## Energy barriers and local versus global stability\n", "\n", "A local minimum may be surrounded by a finite barrier. Perturbations with energy below\n", "the barrier remain in the well under ideal conservative dynamics; larger disturbances\n", "can escape. Local linear stability says nothing about every finite perturbation.\n", "\n", "Report the perturbation size or energy range over which the conclusion is intended.\n", "Parameter uncertainty can shift equilibria and barrier heights, so a stability margin\n", "may be more useful than a single nominal label.\n", "\n", "## Validation checks\n", "\n", "- Confirm the full stationary state, including zero velocity where required.\n", "- Verify force and potential derivative agree.\n", "- Use second derivatives only when their sign test is decisive.\n", "- Examine higher-order terms when curvature vanishes.\n", "- Use Jacobian eigenvalues for velocity-dependent and coupled systems.\n", "- Restrict perturbations to allowed constraint directions.\n", "- Test smaller perturbations and smaller numerical steps independently.\n", "- Distinguish bounded conservative motion from asymptotic decay.\n", "- Report local stability separately from barrier-limited global behavior.\n", "- Preserve units and scaling in gradients, Hessians, and eigenvalue tolerances.\n", "\n", "Equilibrium is a zero of the complete dynamics. Stability is a statement about nearby\n", "trajectories, and it becomes reliable only after the appropriate local model,\n", "constraints, and numerical convergence are all checked.\n", "\n", "## A worked Python notebook\n", "\n", "Inspect how the local curvature changes across a double-well potential. The question for this calculation is precise: how does **double-well 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", "Curvature classifies an equilibrium only after the first derivative has been checked to vanish there. 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", "double-well 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 double-well 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 double-well 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(\"Double-well 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", "double-well 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 double-well 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: Curvature classifies an equilibrium only after the first derivative has been checked to vanish there. 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 8.4, \"Potential Energy Diagrams and Stability\"](https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.4, \"Damped Oscillations\"](https://openstax.org/books/university-physics-volume-1/pages/15-4-damped-oscillations).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 14, Section 14.6, \"Spring Force Energy Diagram\"](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, \"Equilibrium\"](https://physics.info/equilibrium/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "stability-equilibrium-points.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/stability-equilibrium-points", "source_sha256": "d4144dc4fdfeedffe5b0152f39b1efa699aeb42ac59cee1616f0d95ecfd5e366" } }, "nbformat": 4, "nbformat_minor": 5 }