Departure question
Catch change and accumulationPort 9 of 18Follow the languages built to calculate a world that will not stand still, from planets and fluids to waves, optimization, and chaos.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Which path makes the first-order change of a quantity vanish under small variations? This stationarity condition can describe a minimum, maximum, or saddle point. It appears in minimal surfaces, geometrical optics, Lagrangian mechanics, and action principles in relativity, with different admissible paths and boundary conditions in each case.
Problem | Quantity minimized | Solution | Contributor · year |
|---|---|---|---|
Brachistochrone | Descent time | Cycloid | Bernoulli, 1696 |
Shortest path between two points | Distance ∫ds | Straight line (plane) / geodesic (surface) | Euclid |
Largest area for a fixed perimeter | −area | Circle | Queen Dido (legend) |
Shape of a soap film | Area ∫dA | Minimal surface (mean curvature 0) | Plateau, 1849 |
Path of light through a varying refractive index | Stationary optical travel time | Fermat’s principle | Fermat, 1662 |
Planetary orbit | ∫(T − V)dt (action) | Newton’s equations of motion | Hamilton, 1834 |
General relativity | ∫R√−g d⁴x (constants omitted) | Einstein field equations | Einstein and Hilbert, 1915 |
The calculus of variations asks when an entire admissible path, rather than a single number, is stationary under small changes. Additional conditions decide whether a physical solution is actually a minimum.
Optimization where the unknown is itself a function — the foundation of physical "principle of least action".
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Which curve carries a sliding body between two points in the least time? Bernoulli challenged Europe’s mathematicians; the answer is a cycloid.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearNewton solved the brachistochrone problem rapidly and submitted his answer anonymously. Bernoulli reportedly recognized him “by the lion’s claw.”
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearWork by Euler and later Lagrange produced a general method for variational problems, opening a path to expressing physical laws through stationary action.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearLagrange rebuilt mechanics in an analytic form centered on generalized coordinates and variational principles, a language later extended throughout physics.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearLagrangian and Hamiltonian physics, geodesics in general relativity, loss minimization in machine learning, Fermat’s principle in optics, and economics.
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Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Follow the languages built to calculate a world that will not stand still, from planets and fluids to waves, optimization, and chaos.
Open the number voyage
Follow the languages built to calculate a world that will not stand still, from planets and fluids to waves, optimization, and chaos.
Open the number voyage
Follow the languages built to calculate a world that will not stand still, from planets and fluids to waves, optimization, and chaos.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Calculus of Variations
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
