On the brachistochrone problem for cycling ascents

This paper demonstrates that, unlike the classical cycloid solution for gravitational descent, the trajectory that minimizes ascent time for a cyclist with a fixed average power is a straight line with constant grade and speed, implying that steeper hills are optimal for maximizing climbing efficiency (VAM) within physical constraints.

The Gist

Imagine you are a cyclist staring up at a mountain. You have a specific goal: reach the summit (a certain height gain) as fast as possible. You also have a specific engine: your legs, which can only produce a certain amount of average power (energy per second).

The big question is: What is the best path to take?

Should you take a long, winding, gentle path? Or should you go straight up the steepest cliff face you can find?

This paper answers that question with a surprising, almost counter-intuitive result: To get to the top fastest, you should ride the steepest straight line possible.

Here is the breakdown of the paper's logic, translated into everyday language with some helpful analogies.

1. The "Engine" vs. The "Road"

Think of your legs as a car engine with a fixed fuel tank and a fixed horsepower. In physics terms, this is your Average Power.

The paper starts by looking at the forces fighting against you:

• Gravity: Pulling you down.
• Air Resistance: Pushing against you (like running into a wall).
• Friction: The tires rubbing against the road.

The authors prove a mathematical rule: If you have a fixed amount of power to spend, the fastest way to climb is to keep your speed constant. You shouldn't speed up on flat parts and slow down on steep parts. You should just cruise at one steady pace.

2. The "Straight Line" Surprise

Now, imagine you need to get from Point A (the bottom) to Point B (the top). There are infinite paths you could take:

• Path A: A long, winding road that goes around the mountain (long distance, gentle slope).
• Path B: A straight, steep shot up the side of the mountain (short distance, steep slope).

The Classical Trap:
In the famous "Brachistochrone" problem (which is about falling down a hill under gravity), the fastest path is a curved shape called a cycloid. It's like a slide that starts steep to build speed, then flattens out. Gravity helps you speed up, so you want to get that speed early.

The Cycling Reality:
When you are climbing up, gravity is your enemy, not your friend. You are fighting against it the whole time.

• If you take the long, gentle path, you are fighting gravity for a long time. Even though the hill is easier, you are on it for much longer.
• If you take the short, steep path, you fight gravity harder, but you finish the fight much sooner.

The Analogy:
Imagine you are carrying a heavy backpack up a hill.

• The Gentle Path: It's like walking up a long, gentle ramp. It feels easy, but you have to walk for an hour.
• The Steep Path: It's like climbing a ladder. It feels much harder, but you only have to climb for 10 minutes.

The paper proves mathematically that, as long as you have the power to do it, the 10-minute ladder climb is faster than the 1-hour ramp walk. The "straight line" (the shortest distance) is always the winner because it minimizes the time you spend fighting gravity.

3. The "But Wait..." (The Real-World Limits)

If the math says "go straight up a vertical wall," why don't cyclists do that?

The paper adds a crucial "Reality Check." While the math says the steeper the better, physics and biology say there's a limit.

• The Cadence Problem: If the hill is too steep, you have to pedal very slowly (low RPM) to keep moving. This is like trying to run in slow motion while carrying a heavy box; your muscles get inefficient, and you might lose your balance.
• The Wheelie Problem: If the hill is too steep, the front wheel of your bike wants to lift off the ground.
• The Skid Problem: If the hill is too steep, the back wheel might just spin in place without moving you forward.

The "Sweet Spot" Analogy:
Think of your power-to-weight ratio (how strong you are relative to how heavy you are) as a "budget."

• If you are a light, strong rider (a high budget), you can afford to tackle very steep hills (up to about 30% grade). For you, the steeper the better.
• If you are a heavier or weaker rider (a tighter budget), you might not have the power to climb a 30% hill without your speed dropping to a crawl. For you, the "optimal" path might be a slightly less steep hill where you can maintain a faster speed.

The Big Takeaway

The paper concludes that the "Brachistochrone" (the fastest path) for climbing a hill is simply the straight line connecting the start and finish.

• The Ideal: Ride the steepest, straightest line possible.
• The Constraint: You can only go as steep as your bike handling skills and your ability to pedal efficiently allow.

In short: Don't take the scenic, winding route to save energy. If you have the power, cut the corner. Go straight up the steepest hill you can physically manage, keep a steady speed, and you will reach the top faster than anyone taking the long way around.

Technical Summary

Here is a detailed technical summary of the paper "On the brachistochrone problem for cycling ascents" by Bos et al.

1. Problem Statement

The paper addresses the brachistochrone problem specifically for cycling ascents. The objective is to determine the trajectory and riding strategy that minimizes the time required to achieve a specific vertical height gain ($H$), given a fixed average power constraint ($P_0$).

This is equivalent to maximizing VAM (velocit`a ascensionale media), a metric representing the average ascent velocity used to quantify a cyclist's climbing ability. The authors distinguish their work from previous studies by seeking the optimal profile (the shape of the hill) for a given power, rather than optimizing the strategy for a pre-existing hill profile.

2. Methodology

The authors employ a phenomenological physics model combining mechanics and energy constraints.

• Power Equation: They utilize a standard power balance equation (Equation 1) relating power ($P$) to ground speed ($V$), slope ($\theta$), and various resistive forces:
  $$P = \left( m \frac{dV}{dt} + mg \sin \theta + C_{rr} mg \cos \theta + \frac{1}{2} C_d A \rho V^2 \right) \frac{V}{1-\lambda}$$
  Where terms represent acceleration, gravity, rolling resistance, air resistance, and drivetrain efficiency ($\lambda$).

• Optimization Constraints:

  • Constant Speed Assumption: Citing previous work (Bos et al., 2025a), they establish that for a fixed average power, the time-minimizing strategy involves maintaining a constant ground speed ($V$) throughout the ascent.
  • Implicit Speed Function: Under a constant power constraint ($P_0$), the optimal speed $V$ becomes a function of the path length ($L$) and the geometry of the ascent (height $H$ and horizontal distance $D$).

• Mathematical Proof:
  The core of the methodology involves proving two lemmas to demonstrate that the time of ascent ($T$) is a strictly increasing function of the path length ($L$) and the horizontal distance ($D$).

  • Lemma 1: Proves that for fixed start and end points, $dT/dL > 0$. This implies that any deviation from the shortest path (increasing $L$) increases the time required.
  • Lemma 2: Proves that for a fixed height gain $H$, $dT/dD > 0$. This implies that increasing the horizontal distance (making the slope gentler) increases the time required.

3. Key Contributions

• Theoretical Resolution of the Ascent Brachistochrone: The paper provides a rigorous proof that, unlike the classical descent brachistochrone (which is a cycloid), the ascent brachistochrone is a straight line.
• Constant Power/Speed Equivalence: The authors demonstrate that for a cyclist with a fixed average power, the optimal strategy is to climb a straight line at a constant speed, which inherently results in constant instantaneous power.
• Distinction from Classical Mechanics: The paper highlights a fundamental difference between gravity-driven descent (where potential energy converts to kinetic energy, allowing speed to vary) and power-driven ascent (where the cyclist must supply energy, making the shortest path the most efficient).
• Practical Limitations: The paper integrates theoretical physics with practical cycling constraints, identifying that while the mathematical optimum is a vertical climb, physical limits (cadence, balance, traction) impose a maximum feasible grade.

4. Results

• Geometric Optimum: The trajectory of minimum ascent time is the straight line connecting the start and end points.
• Steepness vs. Time:
  • As the slope ($\theta$) increases, the ground speed ($V$) decreases due to the power constraint (gravity and rolling resistance consume more power).
  • However, the vertical component of speed ($V \sin \theta$) increases with steepness.
  • Consequently, the total time to gain a specific height decreases as the slope becomes steeper.
• Numerical Insights:
  • Simulations show that for a fixed power (e.g., 300 W), ascent time decreases monotonically as the slope increases from 5° to 20°.
  • The "30% Limit": While the mathematical limit suggests a vertical climb (90°), practical limitations (pedaling efficiency, front-wheel lift, rear-wheel skid) cap the feasible slope at approximately 30%.
  • Power-to-Weight Ratio: For riders with high power-to-weight ratios (>5 W/kg), the optimal strategy is to ride at the steepest feasible grade (approx. 30%) rather than attempting gentler slopes or unrealistic vertical climbs.

5. Significance

• Strategic Guidance for Cyclists: The findings provide a clear strategic directive: to maximize climbing speed (VAM), cyclists should seek the steepest possible direct route between two points, provided they can maintain traction and pedaling efficiency.
• Clarification of VAM Optimization: The paper resolves ambiguity in climbing strategy by proving that "taking the long way around" a hill (to reduce the grade) is mathematically suboptimal for time minimization under a fixed power budget.
• Contrast with Descent: The work underscores a unique physical distinction in cycling dynamics: while gravity favors a curved path (cycloid) for fastest descent, human-powered ascent favors the shortest geometric path (straight line).
• Model Validation: The study validates the use of constant-speed models for ascent optimization and provides a framework for analyzing the trade-offs between power, mass, and slope in competitive cycling.

In conclusion, the paper establishes that the brachistochrone for cycling ascents is a straight line, and the optimal strategy is to climb the steepest feasible grade at a constant speed, limited only by the cyclist's power-to-weight ratio and biomechanical handling constraints.