{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Phase space and stability\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/phase-space-stability). 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": [ "Phase space represents the complete instantaneous state of a dynamical system. For a\n", "one-dimensional oscillator, position alone is not enough to predict future motion;\n", "position and velocity together are. A phase portrait shows how that state evolves.\n", "\n", "## State instead of position alone\n", "\n", "For a mass-spring oscillator, define state vector\n", "\n", "\n", "$$\n", "\\mathbf y=\\begin{bmatrix}x\\\\v\\end{bmatrix}\n", "$$\n", "\n", "\n", "The state equations are\n", "\n", "\n", "$$\n", "\\dot x=v\\qquad\\dot v=-\\frac{k}{m}x\n", "$$\n", "\n", "\n", "At a given position, the oscillator may be moving left or right. Velocity distinguishes\n", "those futures. Time is a parameter along the curve; a phase plot commonly places\n", "$x$ horizontally and\n", "$v$ vertically." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "output_times = np.linspace(0, 20, 2001)\n", "beta, omega = 0.2, np.sqrt(1 - 0.2**2)\n", "x = np.exp(-beta * output_times) * np.cos(omega * output_times)\n", "v = np.exp(-beta * output_times) * (\n", " -beta * np.cos(omega * output_times) - omega * np.sin(omega * output_times)\n", ")\n", "phase_points = np.column_stack([x, v])\n", "print(\"Final phase point:\", phase_points[-1])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The rendering transform may rescale axes, but it must not modify the saved physical state.\n", "\n", "## Vector field in the phase plane\n", "\n", "At each phase point, the differential equation supplies a tangent vector:\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=\\begin{bmatrix}v\\\\-kx/m\\end{bmatrix}\n", "$$\n", "\n", "\n", "This field says which direction a trajectory moves. At positive displacement with zero\n", "velocity, it points toward negative velocity. At equilibrium with positive velocity, it\n", "points toward increasing position. A trajectory must remain tangent to this field.\n", "\n", "Autonomous deterministic trajectories cannot cross at two different directions in a\n", "phase plane because one state has one derivative. Apparent crossings can result from\n", "projecting a higher-dimensional or explicitly driven system into only two coordinates.\n", "\n", "## Closed curves for ideal harmonic motion\n", "\n", "Mechanical energy is\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+\\frac12kx^2\n", "$$\n", "\n", "\n", "For fixed energy, divide by\n", "$E=\\tfrac12kA^2$ to obtain\n", "\n", "\n", "$$\n", "\\frac{x^2}{A^2}+\\frac{v^2}{\\omega_0^2A^2}=1\n", "$$\n", "\n", "\n", "Each nonzero energy produces a closed ellipse. The origin is the equilibrium state.\n", "Different initial phases choose different starting points on the same energy ellipse.\n", "Different amplitudes choose nested ellipses.\n", "\n", "Because position and velocity use different units, the apparent ellipse aspect ratio\n", "depends on plotting scales. Using normalized vertical coordinate\n", "$v/\\omega_0$\n", "makes the ideal energy curves circles in coordinates with matching length units.\n", "\n", "## Fixed points and local stability\n", "\n", "A fixed point satisfies\n", "\n", "\n", "$$\n", "\\dot x=0\\qquad\\dot v=0\n", "$$\n", "\n", "\n", "For the unforced harmonic oscillator, only\n", "$(x,v)=(0,0)$ qualifies. Nearby\n", "trajectories remain nearby but do not approach the origin, so this conservative center is\n", "stable but not asymptotically stable.\n", "\n", "For a general autonomous system\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=\\mathbf f(\\mathbf y)\n", "$$\n", "\n", "\n", "linearize around fixed point\n", "$\\mathbf y_0$:\n", "\n", "\n", "$$\n", "\\dot{\\boldsymbol\\eta}=\\mathbf J(\\mathbf y_0)\\boldsymbol\\eta\n", "$$\n", "\n", "\n", "where $\\boldsymbol\\eta$ is\n", "the displacement in state space. Eigenvalues of Jacobian\n", "$\\mathbf J$ classify hyperbolic\n", "fixed points: negative real parts decay, a positive real part grows, and opposite signs\n", "form a saddle. Purely imaginary or zero real parts make nonlinear and conservation\n", "structure important.\n", "\n", "## Damping produces an inward spiral\n", "\n", "For linear damping,\n", "\n", "\n", "$$\n", "\\dot x=v\\qquad\\dot v=-\\frac{k}{m}x-\\frac{c}{m}v\n", "$$\n", "\n", "\n", "and energy changes according to\n", "\n", "\n", "$$\n", "\\frac{dE}{dt}=-cv^2\\leq0\n", "$$\n", "\n", "\n", "An underdamped trajectory spirals inward as it crosses equilibrium repeatedly with\n", "decreasing energy. Critical and overdamped trajectories approach the origin without\n", "repeated loops. The origin becomes an attracting fixed point for positive damping.\n", "\n", "An inward spiral in an allegedly undamped calculation instead indicates numerical\n", "dissipation. An outward spiral indicates numerical energy gain or physical negative\n", "damping. Always compare with the energy ledger and step refinement.\n", "\n", "## Unstable equilibria and saddles\n", "\n", "An inverted oscillator with force\n", "$F=+kx$ obeys\n", "\n", "\n", "$$\n", "\\dot x=v\\qquad\\dot v=\\frac{k}{m}x\n", "$$\n", "\n", "\n", "The origin is a saddle. Special initial conditions lie on a stable direction and approach\n", "the fixed point, while generic nearby conditions depart along an unstable direction. A\n", "single exact computation initialized at the origin remains there and therefore cannot\n", "demonstrate stability. Apply small perturbations on both sides.\n", "\n", "Separatrices are trajectories that divide qualitatively different motion. In a nonlinear\n", "pendulum phase portrait, a separatrix divides oscillations from full rotations and passes\n", "through the unstable upright equilibrium. Numerical error near a separatrix can change\n", "the apparent motion class, so energy and refinement are essential.\n", "\n", "## Driven systems need an extended state\n", "\n", "A sinusoidally driven oscillator depends explicitly on time:\n", "\n", "\n", "$$\n", "m\\ddot x+c\\dot x+kx=F_0\\cos(\\omega t)\n", "$$\n", "\n", "\n", "Two identical position-velocity pairs at different drive phases need not have the same\n", "derivative. Add drive phase as a state variable,\n", "\n", "\n", "$$\n", "\\dot\\theta=\\omega\\qquad\\theta\\bmod 2\\pi\n", "$$\n", "\n", "\n", "to make the description autonomous in a higher-dimensional space. A two-dimensional\n", "plot is then only a projection and can appear to cross.\n", "\n", "## Poincare sampling\n", "\n", "For periodic drive, sample once per drive period at one fixed phase. This Poincare section\n", "removes the trivial drive cycle and reveals long-term structure. A settled period-one\n", "response gives one point. A period-two response gives two points. A quasiperiodic response\n", "can form a curve, while chaotic dynamics can give a complicated set." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "drive_frequency = 1.5\n", "period = 2 * np.pi / drive_frequency\n", "sample_times = period * np.arange(1, 11)\n", "# A pure steady sinusoid returns to one phase point each drive period.\n", "poincare = np.column_stack(\n", " [\n", " np.cos(drive_frequency * sample_times),\n", " -drive_frequency * np.sin(drive_frequency * sample_times),\n", " ]\n", ")\n", "assert np.allclose(poincare, [1.0, 0.0])\n", "print(poincare)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Sampling must occur at a consistent physical phase, using interpolation or event location\n", "when internal steps do not land exactly there. A scatter of points caused by timing jitter\n", "is not evidence of chaos.\n", "\n", "## Sampling and visualization choices\n", "\n", "Mark time direction, initial point, and current point. A closed curve without direction\n", "hides whether the state moves clockwise or counterclockwise. Preserve units and avoid\n", "using pixel distance as physical energy.\n", "\n", "Downsampling can skip narrow loops or turning behavior. Connecting widely separated\n", "samples can create lines the trajectory never followed. Retain original state rows and\n", "compare plots under denser output sampling without changing the integration step.\n", "\n", "Phase portraits for several initial conditions should be generated independently but with\n", "identical parameters. They reveal regions of state space, while one trajectory reveals\n", "only its own invariant set.\n", "\n", "## Numerical artifacts and tests\n", "\n", "Ordinary forward Euler makes an ideal harmonic phase ellipse spiral outward. Some methods\n", "spiral inward. Symplectic methods generally keep energy error bounded but can distort\n", "phase. An apparently stable fixed point can be created by excessive numerical damping.\n", "\n", "Use analytic harmonic motion as a fixture. Check that an integer number of periods returns\n", "near the initial state, energy remains within its expected numerical behavior, and the\n", "trace converges as time step shrinks. Test damped energy monotonicity and a known saddle\n", "with signed perturbations.\n", "\n", "For complicated portraits, compare invariant statistics and Poincare structure across\n", "time steps, solvers, durations, and initial perturbations. A phase plot becomes scientific\n", "evidence only when its state definition, sampling, and numerical accuracy are explicit.\n", "\n", "## A worked Python notebook\n", "\n", "Use an exponential envelope to track approach toward a damped oscillator's equilibrium. The question for this calculation is precise: how does **radial envelope in scaled phase space**\n", "change when we change **time**, 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", "Position and velocity require stated scales before a phase-space distance has meaning. 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", "0 to 10 s. 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 time, expressed in s. The output is\n", "radial envelope in scaled phase space, expressed in dimensionless. 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", "$$\n", "f(q) = e^{- 0.2 q}\n", "$$\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 3 s, the expression gives 0.548812 dimensionless. 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(\"exp(-0.2*q)\", locals={\"q\": q})\n", "reference_input = 3.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\n", " \"Predicted radial envelope in scaled phase space:\",\n", " reference_output,\n", " \"dimensionless\",\n", ")\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 dimensionless. The midpoint, at 5 s, gives\n", "0.367879 dimensionless. The upper endpoint gives 0.135335 dimensionless.\n", "Read these together with the assumptions: they describe different choices of time\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([0.0, 5.0, 10.0])\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} s -> {output_value:.6g} dimensionless\")\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 time; the vertical axis shows radial envelope in scaled phase space. 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(0.0, 10.0, 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(\"Time (s)\")\n", "ax.set_ylabel(\"Radial envelope in scaled phase space (dimensionless)\")\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", "radial envelope in scaled phase space change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 3 s is\n", "-0.109762 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", "$$\n", "f'(q) = - 0.2 e^{- 0.2 q}\n", "$$\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-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 3.05\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, \"dimensionless\")\n", "print(\"Full expression:\", full_prediction, \"dimensionless\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 3.05 s. The tangent estimate is\n", "0.543324 dimensionless, while the full expression gives\n", "0.543351 dimensionless. 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 = 10.0\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", "time inside the stated interval, predict whether radial envelope in scaled phase space 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: Position and velocity require stated scales before a phase-space distance has meaning. 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 dimensionless:\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(10.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 10.0, 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*, Sections 15.1-15.2, simple harmonic motion and energy](https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.5, \"Damped Oscillations\"](https://openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.6, \"Forced Oscillations\"](https://openstax.org/books/university-physics-volume-1/pages/15-6-forced-oscillations).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 23, \"Simple Harmonic Motion\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter23/).\n", "- [Benjamin Crowell, *Light and Matter*, oscillation and phase-space discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, \"Phase Plane\"](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/03%3A_One-Dimensional_Motion/3.02%3A_Phase_Plane)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "phase-space-stability.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/phase-space-stability", "source_sha256": "2523040d35214ff81a4dfa663dc8a22a6c268893b037deeead1d9119b277650a" } }, "nbformat": 4, "nbformat_minor": 5 }