{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Phase space visualization\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/phase-space-visualization). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A graph of position against time shows where an object goes. A phase-space graph shows\n", "the state from which its motion continues. For one particle moving along one line, the\n", "state can be represented by position and velocity, so one point\n", "$(x,v)$ contains enough\n", "information to continue a deterministic second-order equation of motion.\n", "\n", "## State space is not ordinary space\n", "\n", "The horizontal coordinate in a phase portrait may be position\n", "$x$, while the vertical coordinate is\n", "velocity $v$. These axes describe\n", "different physical quantities and usually have different units. The distance or angle\n", "drawn on the page therefore has no direct geometric meaning until a plotting scale is\n", "chosen.\n", "\n", "A phase-space trajectory is also not the path of the object through the room. The object\n", "might move back and forth on a straight line while its phase portrait forms a closed\n", "loop. Each point on the loop records one position-velocity pair. Arrows or time coloring\n", "are needed because the same geometric curve can otherwise hide the direction of motion.\n", "\n", "For a system with coordinate $x$ and\n", "momentum $p=mv$,\n", "mechanics often uses the canonical phase plane\n", "$(x,p)$ instead. For\n", "constant mass, the velocity and momentum portraits differ only by a vertical scale.\n", "For generalized coordinates, variable mass matrices, or Hamiltonian mechanics, the\n", "distinction matters.\n", "\n", "## How the differential equation becomes a flow\n", "\n", "Suppose a one-dimensional particle obeys\n", "\n", "\n", "$$\n", "m\\ddot x=F(x,\\dot x,t)\n", "$$\n", "\n", "\n", "Introduce velocity as a separate state variable. The second-order equation becomes two\n", "first-order equations:\n", "\n", "\n", "$$\n", "\\dot x=v,\\qquad \\dot v=\\frac{F(x,v,t)}{m}\n", "$$\n", "\n", "\n", "At every state, the pair\n", "$(\\dot x,\\dot v)$\n", "specifies the local direction of the trajectory. For an autonomous system, where force\n", "has no explicit time dependence, this direction field is fixed on the phase plane.\n", "Trajectories cannot cross at an ordinary point if the differential equation has a unique\n", "solution: a crossing would assign two future motions to the same complete state." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "def derivative(state):\n", " position, velocity = state\n", " return np.array([velocity, -position])\n", "\n", "\n", "print(\"State derivative:\", derivative([1.0, 0.0]))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "If curves appear to cross in a two-dimensional projection of a higher-dimensional\n", "system, the hidden coordinates can distinguish the states. A time-dependent force also\n", "requires time or drive phase to make the state autonomous.\n", "\n", "## Simple harmonic oscillator\n", "\n", "For an ideal mass-spring oscillator,\n", "\n", "\n", "$$\n", "m\\ddot x=-kx\n", "$$\n", "\n", "\n", "the state equations are\n", "\n", "\n", "$$\n", "\\dot x=v,\\qquad \\dot v=-\\omega^2x,\\qquad \\omega=\\sqrt{\\frac{k}{m}}\n", "$$\n", "\n", "\n", "The analytic solution can be written as\n", "\n", "\n", "$$\n", "x=A\\cos(\\omega t+\\phi),\\qquad v=-A\\omega\\sin(\\omega t+\\phi)\n", "$$\n", "\n", "\n", "Eliminating time gives\n", "\n", "\n", "$$\n", "\\frac{x^2}{A^2}+\\frac{v^2}{A^2\\omega^2}=1\n", "$$\n", "\n", "\n", "Thus each nonzero-energy motion is an ellipse in the\n", "$(x,v)$ plot. In canonical\n", "$(x,p)$ coordinates it is\n", "also an ellipse. If the vertical coordinate is rescaled to\n", "$v/\\omega$,\n", "the ideal contour becomes a circle. Whether a plot looks circular is therefore a choice\n", "of scale, not a new physical law.\n", "\n", "Read Figure 1 by matching the labeled spring positions to points on the ellipse.\n", "At A, the mass is at its right turning point: velocity is zero and acceleration points\n", "left, so the phase-space state initially moves downward. At B, the mass crosses\n", "equilibrium moving left; its negative velocity places the state at the bottom of the\n", "ellipse. At C, the mass reaches its left turning point. Following A to B to C makes\n", "the clockwise direction visible, although the physical mass only moves along a line.\n", "\n", "## Energy contours\n", "\n", "The oscillator's mechanical energy is\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+\\frac12kx^2\n", "$$\n", "\n", "\n", "A conservative trajectory remains on one contour of constant energy. The intercepts\n", "have immediate meanings:\n", "\n", "\n", "$$\n", "x_{\\max}=\\sqrt{\\frac{2E}{k}},\\qquad |v|_{\\max}=\\sqrt{\\frac{2E}{m}}\n", "$$\n", "\n", "\n", "In Figure 1, A and C store all the oscillator's energy in the spring, while B stores\n", "it as kinetic energy. Those states lie on the same ellipse because total energy is\n", "unchanged. Its horizontal and vertical intercepts therefore provide two independent\n", "ways to recover that energy from position and speed.\n", "\n", "This makes a phase portrait useful for checking data. A point outside the expected\n", "contour has too much computed energy; one inside has too little. The conclusion assumes\n", "that mass and spring constant are fixed and that the model includes every relevant\n", "energy transfer.\n", "\n", "For a general one-dimensional conservative system,\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+U(x)\n", "$$\n", "\n", "\n", "so a fixed-energy curve has two velocity branches:\n", "\n", "\n", "$$\n", "v=\\pm\\sqrt{\\frac{2}{m}\\left[E-U(x)\\right]}\n", "$$\n", "\n", "\n", "The curve exists only where\n", "$E\\ge U(x)$.\n", "Points where $E=U(x)$ are\n", "turning points unless they are equilibria approached only asymptotically. Closed curves\n", "usually represent bounded oscillation; open curves can represent escape or transit." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "positions = np.linspace(-1, 1, 201)\n", "speeds = np.sqrt(np.maximum(0, 1 - positions**2))\n", "fig, ax = plt.subplots()\n", "ax.plot(positions, speeds, label=\"Positive velocity\")\n", "ax.plot(positions, -speeds, label=\"Negative velocity\")\n", "ax.set(xlabel=\"Position (m)\", ylabel=\"Velocity (m/s)\")\n", "ax.legend()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Near a separatrix, small changes in energy can change the qualitative type of motion.\n", "For example, in a double-well potential the separatrix distinguishes motion trapped in\n", "one well from motion able to cross the central barrier. Numerical and measurement\n", "uncertainty should be shown when deciding on which side of this boundary a state lies.\n", "\n", "## Equilibria in the portrait\n", "\n", "An equilibrium is a fixed point of the phase flow. For the mechanical system above it\n", "requires both\n", "$v=0$ and\n", "$F(x,0)=0$. A stable\n", "conservative equilibrium is often surrounded by nested closed curves. An unstable\n", "equilibrium often has trajectories that approach along one direction and depart along\n", "another, forming saddle-like geometry.\n", "\n", "Do not classify stability from a rough plot alone. Curves that seem to spiral may simply\n", "be dense closed contours or plotting artifacts. Confirm the fixed point analytically,\n", "linearize the equations, and use a resolution study if data were calculated numerically.\n", "\n", "## Damping and driving\n", "\n", "For linear viscous damping,\n", "\n", "\n", "$$\n", "m\\ddot x+b\\dot x+kx=0\n", "$$\n", "\n", "\n", "the mechanical energy changes at the rate\n", "\n", "\n", "$$\n", "\\frac{dE}{dt}=-bv^2\n", "$$\n", "\n", "\n", "With $b>0$, energy never\n", "increases, so an underdamped phase trajectory spirals inward toward the origin. Critical\n", "and overdamping can approach the origin without making repeated loops. An outward spiral\n", "means energy is entering the modeled mechanical subsystem; it may indicate negative\n", "damping, external driving, or numerical error.\n", "\n", "A periodically driven oscillator has an explicitly time-dependent state rule. Its\n", "two-dimensional traces can overlap because drive phase is hidden. Adding phase as a\n", "state coordinate, or sampling once per drive period to make a Poincare section, restores\n", "a more faithful description. A closed loop in the two-dimensional projection does not\n", "by itself prove energy conservation.\n", "\n", "## Building a trustworthy portrait from data\n", "\n", "Plot measured velocity when it is available. Estimating velocity from noisy positions by\n", "finite differences amplifies high-frequency noise. Smoothing can reduce that noise but\n", "can also erase real fast motion and alter phase. Report the differentiation method and\n", "its time scale.\n", "\n", "Samples must remain in chronological order. A scatter plot alone can show the occupied\n", "region but not the orbit direction or repeated visits. Connect neighboring times only\n", "when gaps are small enough that the connecting segment is meaningful. Mark the initial\n", "state and add sparse arrowheads rather than making every point visually dominant." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "times = np.linspace(0, 2 * np.pi, 201)\n", "positions, velocities = np.cos(times), -np.sin(times)\n", "fig, ax = plt.subplots()\n", "ax.plot(positions, velocities)\n", "ax.scatter([positions[0]], [velocities[0]], label=\"Start\")\n", "ax.annotate(\n", " \"\",\n", " xy=(positions[10], velocities[10]),\n", " xytext=(positions[0], velocities[0]),\n", " arrowprops={\"arrowstyle\": \"->\"},\n", ")\n", "ax.set(xlabel=\"Position (m)\", ylabel=\"Velocity (m/s)\")\n", "ax.legend()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Use equal physical meaning, not necessarily equal pixels, when choosing axis scales.\n", "State the scale explicitly. If several trajectories are compared, keep consistent limits\n", "or explain changes; auto-scaling can make shrinking and growing motions look identical.\n", "\n", "## Numerical method diagnostics\n", "\n", "A phase portrait can expose integration error, but it cannot identify the cause without\n", "additional checks. Forward Euler applied to an ideal oscillator generally produces\n", "artificial outward energy growth. Some dissipative algorithms can spiral inward even\n", "when the true system is conservative. Higher formal order is not the same as preserving\n", "geometric structure over long times.\n", "\n", "Repeat the same initial-value problem with smaller time steps. Compare states at the same\n", "physical times, compare energy error, and examine whether qualitative features such as\n", "closed contours and turning points converge. A symplectic method is often valuable for\n", "long conservative trajectories because it tends to preserve phase-space structure and\n", "keeps energy error bounded, although it does not make every step exact." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "for steps in [100, 200, 400]:\n", " state = np.array([1.0, 0.0])\n", " dt = 6.0 / steps\n", " energy_errors = []\n", " for _ in range(steps):\n", " state += dt * derivative(state)\n", " energy_errors.append(abs(0.5 * np.dot(state, state) - 0.5))\n", " print(f\"dt={dt:.3f}: max energy error {max(energy_errors):.6f} J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "## Dimensions beyond one coordinate\n", "\n", "A system with $n$ independent\n", "configuration coordinates generally needs a phase space with\n", "$2n$: one coordinate and\n", "one conjugate momentum for each degree of freedom. Constraints can reduce that count.\n", "A two-dimensional plot is then a projection, section, or marginal view rather than the\n", "complete state.\n", "\n", "Always name what has been omitted. Two projected trajectories may cross even though the\n", "full phase-space trajectories do not. A useful projection preserves the variables tied\n", "to the question; a useful section states the condition under which points were sampled.\n", "\n", "## Interpretation checks\n", "\n", "- Label axes with variable names and units; never treat their drawn angle as invariant.\n", "- Mark chronological direction and the initial state.\n", "- Confirm that the plotted variables form a complete state for the stated model.\n", "- Check constant-energy contours only for a closed conservative subsystem.\n", "- Interpret inward or outward spirals through an explicit energy balance.\n", "- Distinguish true separatrices from coarse sampling and numerical drift.\n", "- Reduce both integration step and output-sampling interval when checking convergence.\n", "- Treat a low-dimensional display as a projection when hidden state variables exist.\n", "- Compare analytic turning points and equilibria with the plotted curve.\n", "- Keep measured uncertainty and numerical error conceptually separate.\n", "\n", "Phase space turns motion into geometry, but the geometry is meaningful only when the\n", "state variables, units, time direction, and governing equations are kept visible. Used\n", "with energy and convergence checks, it reveals bounded motion, dissipation, instability,\n", "and hidden modeling errors more clearly than a position-time graph alone.\n", "\n", "## A worked Python notebook\n", "\n", "Trace the position component of a unit-frequency harmonic orbit. The question for this calculation is precise: how does **position**\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", "A phase portrait must pair this position with minus sine of the same time as its velocity. 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 6.28319 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", "position, expressed in 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", "$$\n", "f(q) = \\cos{\\left(q \\right)}\n", "$$\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 1.0472 s, the expression gives 0.5 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-011", "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(\"cos(q)\", locals={\"q\": q})\n", "reference_input = 1.0471975511965979\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\"Predicted position:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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 m. The midpoint, at 3.14159 s, gives\n", "-1 m. The upper endpoint gives 1 m.\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-013", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 3.141592653589793, 6.283185307179586])\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} m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "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 position. 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-015", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 6.283185307179586, 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(\"Position (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-016", "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", "position change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1.0472 s is\n", "-0.866025 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", "$$\n", "f'(q) = - \\sin{\\left(q \\right)}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.0786134777324958\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, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 1.07861 s. The tangent estimate is\n", "0.472793 m, while the full expression gives\n", "0.472551 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-019", "metadata": {}, "source": [ "input_span = 6.283185307179586\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-020", "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 position 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 phase portrait must pair this position with minus sine of the same time as its velocity. 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 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-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(6.283185307179586) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 6.283185307179586, 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-022", "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 15.1, \"Simple Harmonic Motion\"](https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.5, \"Forced Oscillations\"](https://openstax.org/books/university-physics-volume-1/pages/15-5-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", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 14, \"Potential Energy and Conservation of Energy\"](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, \"Simple Harmonic Oscillator\"](https://physics.info/sho/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "phase-space-visualization.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/phase-space-visualization", "source_sha256": "4e277fc0f33b80b39c325b47c78684f531509b69dd6c4328bc073c52e0b432e4" } }, "nbformat": 4, "nbformat_minor": 5 }