Brachistochrone Subway Tunnel: Fastest Tunnel Shape Explained
Table of Contents
Introduction
A subway tunnel could, in principle, operate without external energy (like electricity) if we assume zero friction everywhere. The train would descend, convert potential energy to kinetic energy, coast through a low point, then climb back up—all by exchanging potential and kinetic energy.
Historical background
What is the tunnel shape that minimizes travel time between two stations separated by a horizontal distance d? This is the Brachistochrone problem, first posed by Johann Bernoulli in the late 17th century as “what path gives the shortest time for an object to slide along a smooth curve from point A to point B under the influence of gravity?” Bernoulli, his brother, Leibniz, Newton, and two others submitted solutions—the correct curve is the cycloid. (The last entry was submitted anonymously but was later recognized as Newton’s.) Later, Euler and Lagrange generalized the theory; for more historical info see
http://en.wikipedia.org/wiki/Brachistochrone_curve
Problem setup
For a subway we want to minimize the station-to-station travel time. We can apply the Brachistochrone solution to half the journey (from the first station down to the tunnel bottom) and then double the time to get the full transit time. The only fixed parameter is d, the above-ground horizontal distance between the stations.
Set up an x–y coordinate system with x along the ground and y the depth below ground. The train starts at x = 0, y = 0 and ends at x = d, y = 0. Let the maximum depth be h; define h > 0.
Variational approach and equations
The Brachistochrone problem is a special case in the calculus of variations. It extremalizes a definite integral of the form
The integral to be minimized is
Here ds is the element of arc length and v the speed along the curve. Since ds2 = dx2 + dy2 and v is determined from v2/2 = gy, we obtain
f = {[1 + (dy/dx)2]/2gy}1/2.
The corresponding Euler equation takes the form
y’ ∂f/∂y’ – f = c with y’ = dy/dx
In this presentation I omit the intermediate derivation steps, but the integration leads to the cycloid as the time-minimizing path. (I can provide derivation links or references on request.)
Optimal depth and travel time
I calculated the maximum depth required for the minimum station-to-station transit time. The result is
h = d/π, with the corresponding transit time
T = √(2πd/g).
For example, with d = 1 mile (given here as 1600 m in the original calculation), this gives h = 0.32 miles or 1,681 ft (488 m) and T = 32.0 s. That is quite deep and steep: the maximum speed at the bottom would be roughly 225 mph and the above-ground average speed about 112 mph.
Of course, air resistance and rail friction would reduce these speeds and would require externally supplied energy in practice, but the idealized values give a useful baseline.
Shallower tunnels and repeated cycloid segments
If a shallower maximum depth is desired—for example h ≈ 800 ft (244 m)—one option is to preserve the cycloidal shape but shorten the cycloid’s spatial period. In that case the train would descend to 800 ft, ascend back up at d = 1/2 mi, then repeat the cycloidal arc for the second half of the mile. Transit time scales with the square root of the above-ground distance per cycloid period, so the 1-mile time for that two-cycloid arrangement increases to T ≈ 45.2 s.
Alternative: circular-arc tunnel (no intermediate pop-up)
Another design constraint is to avoid popping back to the surface mid-distance. I considered an arc of a circle spanning x = 0 to x = d with the arc bottom at 244 m below the x-axis. That circle is centered at (d/2, R − h), with
$$R = radius = (d/2)^2/8h + h/2$$
The transit-time expression for a circular arc involves elliptic integrals (“elliptical integral of the first kind with parameter m = k^2”) and does not yield a simple closed form in elementary functions, so I evaluated transit times numerically.
Numerical results (Excel simulations)
- d = 1 mile, depth = 244 m: T = 38.7 s (this beats the two-cycloid version at 45.2 s).
- d = 1 mile, h = 100 m: T = 57.2 s.
- d = 1 mile, h = 50 m: T = 80.0 s.
- d = 1 mile, h = 25 m: T = 112.7 s.
- d = 1 mile, h = 10 m: T = 177.3 s.
Conclusion
These results show that a shallow tunnel does not necessarily give acceptable transit times for fast travel, and that the ideal brachistochrone solution (a single deep cycloid) is not likely cost-effective for a real subway system. A compromise—shallower circular arcs or repeated cycloidal segments—can reduce maximum depth while keeping transit time within a useful range.
AB Engineering and Applied Physics
MSEE
Aerospace electronics career
Used to hike; classical music, esp. contemporary; Agatha Christie mysteries.
Interesting topic.
“Where is the point in the circle arcs?
You know the fastest way to get from A to B in general – let the deepest point of the brachistrochrone be at a depth of h. What is the ideal shape if you want to take a longer distance, but with depth limited to h? Well, just add a straight section in between the brachistrochrone halves. It will run at the maximal possible speed, and you already know the first and last part cannot be faster.
For the 1600m distance, you need 22.6 s for the curved sections, and 12 seconds for 800m at 66m/s, for a total of 34.6 seconds.”
Good observation. Thanks.
Where is the point in the circle arcs?
You know the fastest way to get from A to B in general – let the deepest point of the brachistrochrone be at a depth of h. What is the ideal shape if you want to take a longer distance, but with depth limited to h? Well, just add a straight section in between the brachistrochrone halves. It will run at the maximal possible speed, and you already know the first and last part cannot be faster.
For the 1600m distance, you need 22.6 s for the curved sections, and 12 seconds for 800m at 66m/s, for a total of 34.6 seconds.
Following mfb’s observation, the title of my blog should be changed to “Modified Brachistochrone Subway!”. mfb has come up with the ideal solution and it resurrects the viability of the Brachistochrone in subway engineering!
Very unique entry!
Well, this was worth reading. I learned something interesting today, so thanks for writing this.