--- title: "From Coordinates to Fields: the Lagrangian Density" description: "A field has a value at every point, so the sum over coordinates becomes an integral of a Lagrangian density and the equations of motion become partial differential equations. The vibrating string is t" canonical: https://lightmysky.com/learn/science/from-coordinates-to-fields-the-lagrangian-density-mt_fRZN3ZByjr source: https://lightmysky.com/learn/science/from-coordinates-to-fields-the-lagrangian-density-mt_fRZN3ZByjr.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # From Coordinates to Fields: the Lagrangian Density A field has a value at every point, so the sum over coordinates becomes an integral of a Lagrangian density and the equations of motion become partial differential equations. The vibrating string is the case where the step can be checked against an answer already known. Subject: Science · Area: Forces & Motion · Ages 22 to 24 Page: https://lightmysky.com/learn/science/from-coordinates-to-fields-the-lagrangian-density-mt_fRZN3ZByjr ## Ready when they can - Writes a Lagrangian density for a continuous system and derives its field equation - Explains what replaces the index labelling a coordinate when the system becomes a field - Recovers the wave equation from a field Lagrangian - Finds arrival time of a spherical wave front from distance divided by wave speed ## Lesson: From beads on a string to the field itself A field assigns a value to every point in space, so there are infinitely many parts and the label that numbered coordinates is replaced by the position itself. Where you summed over coordinates, you now integrate a Lagrangian density over space: L(t) = ∫ ℒ dx, and the action is S = ∫ dt ∫ ℒ dx. The field equation comes from demanding that S be stationary under bends of the field u(x, t) that vanish at the spatial endpoints and at the initial and final times, or have compact support. For one field the condition reads ∂ℒ/∂u − ∂/∂t(∂ℒ/∂u_t) − ∂/∂x(∂ℒ/∂u_x) = 0. A stretched string with mass per length μ > 0, tension T > 0, and small transverse displacement u(x, t) has density ℒ = ½μ u_t² − ½T u_x². The assumptions are a uniform ideal string, small slopes so u_x stays tiny, and no external force. Then ∂ℒ/∂u = 0, ∂ℒ/∂u_t = μ u_t, and ∂ℒ/∂u_x = −T u_x, so the field equation is μ u_tt − T u_xx = 0. This is the wave equation with wave speed c = √(T/μ). *(drawing: From field to action to the Euler-Lagrange condition.)* u_tt = c² u_xx. The travelling-wave solution is u(x, t) = F(x − ct) + G(x + ct), for any twice differentiable F and G. F(x − ct) travels right at speed c and G(x + ct) travels left at speed c. Released from rest with initial displacement h(x), differentiate: u_t = −cF′(x − ct) + cG′(x + ct), so zero velocity at t = 0 gives F′(x) = G′(x), hence F = G + C. F + G = h gives F = h/2 + C/2 and G = h/2 − C/2, and the constants cancel. u(x, t) = ½h(x − ct) + ½h(x + ct). A string of length L fixed at both ends admits sine shapes that vanish at the ends. Those are sin(nπx/L) with whole numbers n, since each equals zero at x = 0 and x = L. Standing-wave solutions are A sin(nπx/L) cos(ωₙt + φ) with ωₙ = ncπ/L. A static nonzero sin shape alone is not a solution, since its second time derivative vanishes but its second space derivative does not. Always extend the initial displacement outside as an odd function about both endpoints with period 2L, and extend the initial velocity the same way, or set it to zero when the string is released from rest. *(drawing: Fixed-end modes and the odd extension method.)* Formal check: with c = 2 and h(x) = x², u(1, 1) = ½(1 − 2)² + ½(1 + 2)² = ½(1 + 9) = 5. This polynomial example is a formal nondimensional check of the formula on the whole line, not a physical small-slope string, since x² has unbounded slope. A single time-dependent standing mode is an allowed restriction of the full field, not a general replacement for the partial differential equation. In three dimensions a point push in a uniform isotropic medium spreads as a spherical front at constant speed c with no obstacles, so arrival time is distance over speed. **Recap.** Replace the coordinate list with a field, integrate the density, derive μ u_tt − T u_xx = 0 with c = √(T/μ), and check the string gives traveling waves. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Noether's Theorem: Symmetry and Conserved Quantities](https://lightmysky.com/learn/science/noethers-theorem-symmetry-and-conserved-quantities-mt_kAoOn8syBK) - [The Wave Equation and Its Travelling-Wave Solutions](https://lightmysky.com/learn/science/the-wave-equation-and-its-travelling-wave-solutions-mt_Ue-v1cSxNx) ## Optional connections - [The Wave Equation and Its Characteristics](https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w) ## Opens up - [Field Quantisation: Creation Operators and Particle Number](https://lightmysky.com/learn/science/field-quantisation-creation-operators-and-particle-number-mt_BJyZWQA5cb)