{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Symbolic energy methods\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/symbolic-energy-methods). 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": [ "Symbolic energy methods manipulate exact expressions before numbers are substituted.\n", "They can derive forces from potentials, solve for turning points, expose parameter\n", "scaling, and prove conservation under stated equations. Their reliability depends on\n", "units, domains, assumptions, and physical branch selection—not only algebraic equality.\n", "\n", "## Build the energy expression from the model\n", "\n", "For a particle of constant mass in a time-independent potential,\n", "\n", "\n", "$$\n", "E(x,v)=\\frac12mv^2+U(x)\n", "$$\n", "\n", "\n", "Before simplifying, state\n", "$m>0$, the domain of\n", "$x$, the reference frame for\n", "$v$, and the zero convention for\n", "$U$. A symbolic expression without those\n", "assumptions can produce branches that are mathematically valid but physically\n", "irrelevant.\n", "\n", "For a spring in uniform gravity,\n", "\n", "\n", "$$\n", "E=\\frac12mv^2+\\frac12k(x-x_0)^2+mgx\n", "$$\n", "\n", "\n", "The coordinate direction and spring reference\n", "$x_0$ must be defined before\n", "signs can be checked.\n", "\n", "## Derive force from potential\n", "\n", "In one dimension,\n", "\n", "\n", "$$\n", "F(x)=-\\frac{dU}{dx}\n", "$$\n", "\n", "\n", "In multiple Cartesian dimensions,\n", "\n", "\n", "$$\n", "\\mathbf F(\\mathbf r)=-\\nabla U(\\mathbf r)\n", "$$\n", "\n", "\n", "For the ideal spring potential\n", "$U=\\tfrac12kx^2$,\n", "differentiation gives\n", "$F=-kx$. The negative sign is\n", "physical: force points toward lower potential." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import sympy as sp\n", "\n", "x, v = sp.symbols(\"x v\", real=True)\n", "m, k = sp.symbols(\"m k\", positive=True)\n", "potential = k * x**2 / 2\n", "force = -sp.diff(potential, x)\n", "print(\"Force:\", force)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Differentiation should preserve excluded points. For\n", "$U=-GMm/r$, the physical domain\n", "excludes $r=0$ even if a later expression\n", "can be algebraically rearranged there.\n", "\n", "## Prove conservation along the equations of motion\n", "\n", "Differentiate the energy along a trajectory:\n", "\n", "\n", "$$\n", "\\frac{dE}{dt}=mv\\dot v+\\frac{dU}{dx}\\dot x\n", "$$\n", "\n", "\n", "Using $\\dot x=v$ and\n", "$m\\dot v=F=-dU/dx$,\n", "\n", "\n", "$$\n", "\\frac{dE}{dt}=v\\left(m\\dot v+\\frac{dU}{dx}\\right)=0\n", "$$\n", "\n", "\n", "This proof depends on constant mass, a time-independent differentiable potential, and no\n", "other work channels. A computer algebra system can substitute the equation of motion and\n", "simplify to zero, but it cannot infer that omitted friction should have been present." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "energy = m * v**2 / 2 + potential\n", "energy_rate = sp.diff(energy, x) * v + sp.diff(energy, v) * (force / m)\n", "assert sp.simplify(energy_rate) == 0\n", "print(\"Energy rate:\", sp.simplify(energy_rate))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "## Explicit time dependence\n", "\n", "For $U(x,t)$ and conservative\n", "spatial force $F=-\\partial U/\\partial x$,\n", "\n", "\n", "$$\n", "\\frac{d}{dt}(K+U)=\\frac{\\partial U}{\\partial t}\n", "$$\n", "\n", "\n", "The spatial force can still come from the potential while mechanical energy is not\n", "constant. Symbolic code must distinguish partial time derivative from the total\n", "derivative along motion.\n", "\n", "## Solve for speed from energy\n", "\n", "If total energy is known,\n", "\n", "\n", "$$\n", "v^2=\\frac{2}{m}[E-U(x)]\n", "$$\n", "\n", "\n", "and\n", "\n", "\n", "$$\n", "v=\\pm\\sqrt{\\frac{2}{m}[E-U(x)]}\n", "$$\n", "\n", "\n", "The plus-minus matters. Energy determines speed but not direction. The radicand must be\n", "nonnegative in classical motion:\n", "\n", "\n", "$$\n", "E\\ge U(x)\n", "$$" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "\n", "total_energy, position, mass, stiffness = 2.0, 0.5, 1.0, 4.0\n", "available_kinetic = total_energy - 0.5 * stiffness * position**2\n", "if available_kinetic < 0:\n", " raise ValueError(\"This position is classically forbidden at the chosen energy.\")\n", "speed = np.sqrt(2 * available_kinetic / mass)\n", "print(\"Allowed velocities (m/s):\", -speed, speed)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Clamping only a tiny negative value consistent with roundoff is different from erasing a\n", "genuinely forbidden state.\n", "\n", "## Turning points and equilibrium points\n", "\n", "A turning point for one-dimensional motion with total energy\n", "$E$ satisfies\n", "\n", "\n", "$$\n", "U(x_{\\mathrm{turn}})=E\n", "$$\n", "\n", "\n", "because kinetic energy and instantaneous speed are zero there. Not every root is reached\n", "from a chosen initial condition; disconnected allowed regions and intervening barriers\n", "matter.\n", "\n", "Equilibria satisfy\n", "\n", "\n", "$$\n", "\\frac{dU}{dx}=0\n", "$$\n", "\n", "\n", "For an isolated stationary point,\n", "$d^2U/dx^2>0$\n", "indicates a local minimum and stable small displacements in the ideal conservative\n", "model, while a negative second derivative indicates a local maximum and instability.\n", "A zero second derivative requires higher-order analysis.\n", "\n", "## Example: vertical launch by energy\n", "\n", "For an object launched upward in uniform gravity, choose zero potential at launch. Then\n", "\n", "\n", "$$\n", "E=\\frac12mv_0^2=\\frac12mv^2+mgy\n", "$$\n", "\n", "\n", "At the highest point,\n", "$v=0$, so\n", "\n", "\n", "$$\n", "y_{\\max}=\\frac{v_0^2}{2g}\n", "$$\n", "\n", "\n", "Mass cancels, consistent with uniform-gravity kinematics. This result neglects drag and\n", "assumes the gravitational field remains uniform. Symbolic cancellation does not make\n", "those assumptions universal.\n", "\n", "## Example: shifted spring equilibrium\n", "\n", "For\n", "\n", "\n", "$$\n", "U(x)=\\frac12kx^2+mgx\n", "$$\n", "\n", "\n", "complete the square:\n", "\n", "\n", "$$\n", "U(x)=\\frac12k\\left(x+\\frac{mg}{k}\\right)^2-\\frac{m^2g^2}{2k}\n", "$$\n", "\n", "\n", "The equilibrium is shifted to\n", "$x_{\\mathrm{eq}}=-mg/k$.\n", "The final constant changes the chosen potential zero but not force or oscillation\n", "frequency. Completing the square exposes the physics more clearly than expanding every\n", "term numerically.\n", "\n", "## Symbolic dimensional checks\n", "\n", "Attach dimensions to symbols before manipulating expressions. Terms joined by addition\n", "must share dimensions. In\n", "\n", "\n", "$$\n", "\\frac12mv^2+mgh\n", "$$\n", "\n", "\n", "both terms have energy dimensions. The argument of an exponential, logarithm, sine, or\n", "cosine must be dimensionless; a symbolic expression such as\n", "$\\cos(\\omega t)$ is valid because\n", "angular frequency times time is dimensionless." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "# Exponents of mass, length, time for each energy term.\n", "kinetic_dimension = (1, 2, -2)\n", "spring_dimension = (1, 0, -2)\n", "position_squared_dimension = (0, 2, 0)\n", "potential_dimension = tuple(\n", " a + b for a, b in zip(spring_dimension, position_squared_dimension)\n", ")\n", "assert kinetic_dimension == potential_dimension\n", "print(\"Both terms have energy dimensions.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "A computer algebra system without unit metadata can simplify a dimensionally impossible\n", "sum as readily as a correct one.\n", "\n", "## Assumptions during simplification\n", "\n", "The identity\n", "$\\sqrt{x^2}=|x|$\n", "does not simplify to $x$ unless\n", "$x\\ge0$ is known. Dividing by a symbol\n", "assumes it is nonzero. Squaring both sides can introduce extra roots. Taking a square\n", "root discards a sign unless both branches are restored.\n", "\n", "For energy equations these details determine direction, turning points, and allowed\n", "domains. Keep the original equation and filter candidate solutions by all initial\n", "assumptions.\n", "\n", "## Piecewise and nonsmooth potentials\n", "\n", "Absolute-value, contact, and piecewise potentials may not be differentiable at their\n", "joins. A symbolic derivative can contain piecewise expressions or distributions. A\n", "single formula evaluated exactly at the corner may not define a unique ordinary force.\n", "\n", "State the left and right forces, event rule, or finite compliant approximation. Do not\n", "smooth a corner silently merely to make differentiation convenient; smoothing defines a\n", "new model with a scale parameter.\n", "\n", "## Use exact expressions as test oracles\n", "\n", "Symbolic results can generate numerical fixtures:\n", "\n", "- compare a coded spring force with the exact derivative;\n", "- compare integrated conservative work with endpoint potential difference;\n", "- substitute an analytic trajectory into the energy expression;\n", "- test turning-point roots by substitution into the original equation;\n", "- evaluate limiting cases such as zero gravity or zero spring displacement;\n", "- compare a series approximation with the exact expression over a stated range." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "turning_points = sp.solve(sp.Eq(potential.subs(k, 4), 2), x)\n", "for root in turning_points:\n", " assert sp.simplify(potential.subs({k: 4, x: root}) - 2) == 0\n", "print(\"Turning points (m):\", turning_points)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "## Numerical verification remains necessary\n", "\n", "Equivalent symbolic expressions can behave differently in floating-point arithmetic.\n", "Subtracting nearly equal large energies can lose significant digits. Factored forms can\n", "avoid overflow or underflow in one range and worsen it in another. Test generated code\n", "with high-precision references, boundary values, and representative scales.\n", "\n", "Symbolic proof that\n", "$dE/dt=0$ applies to continuous\n", "equations. A finite-step integrator generally does not conserve that exact expression\n", "at every step. Symbolic and numerical claims must not be conflated.\n", "\n", "## Checks for a symbolic energy workflow\n", "\n", "- State symbol domains, units, frame, and potential zero.\n", "- Derive force from negative gradient and compare with the implemented force.\n", "- Preserve excluded singular points during simplification.\n", "- Keep both velocity branches after solving an energy square.\n", "- Substitute candidate roots into the original equation and physical domain.\n", "- Distinguish partial and total time derivatives.\n", "- Test equilibria with higher derivatives when the second derivative vanishes.\n", "- Handle piecewise joins and nondifferentiable points explicitly.\n", "- Verify symbolic results numerically at ordinary, limiting, and boundary cases.\n", "- Keep continuous conservation proofs separate from discrete integrator behavior.\n", "\n", "Symbolic energy methods reveal structure before arithmetic hides it. Their best role is\n", "to connect potentials, forces, allowed motion, and conservation while carrying the\n", "physical assumptions that make each algebraic step valid.\n", "\n", "## A worked Python notebook\n", "\n", "Start with a spring potential and derive its slope symbolically. The question for this calculation is precise: how does **potential energy**\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", "The physical spring force is the negative of this slope, not the slope itself. 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", "-2 to 2 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", "potential energy, expressed in J. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 2 q^{2}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 1 m, the expression gives 2 J. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-013", "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(\"2*q**2\", 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 potential energy:\", reference_output, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "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", "8 J. The midpoint, at 0 m, gives\n", "0 J. The upper endpoint gives 8 J.\n", "Read these together with the assumptions: they describe different choices of position\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-2.0, 0.0, 2.0])\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} m -> {output_value:.6g} J\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "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 potential energy. 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-017", "metadata": {}, "source": [ "input_grid = np.linspace(-2.0, 2.0, 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(\"Position (m)\")\n", "ax.set_ylabel(\"Potential energy (J)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "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", "potential energy 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", "4 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", "$$\n", "f'(q) = 4 q\n", "$$\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-019", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.02\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"J\")\n", "print(\"Full expression:\", full_prediction, \"J\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "Here the changed input is 1.02 m. The tangent estimate is\n", "2.08 J, while the full expression gives\n", "2.0808 J. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-021", "metadata": {}, "source": [ "input_span = 4.0\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-022", "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 potential energy 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: The physical spring force is the negative of this slope, not the slope itself. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of J:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-023", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(2.0) - evaluate(-2.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-2.0, 2.0, 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-024", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 8.1, \"Potential Energy of a System\"](https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system).\n", "- [OpenStax, *University Physics Volume 1*, Section 8.2, \"Conservative and Non-Conservative Forces\"](https://openstax.org/books/university-physics-volume-1/pages/8-2-conservative-and-non-conservative-forces).\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, \"Potential Energy\"](https://physics.info/energy-potential/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "symbolic-energy-methods.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/symbolic-energy-methods", "source_sha256": "4c41d408da17a0e3a8c7435aa8edd2b8495d48d49434c9cccfb337e1961ec0c1" } }, "nbformat": 4, "nbformat_minor": 5 }