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➗calculus ii review

Brachistochrone Curve

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The brachistochrone curve is the shape of the path that allows a frictionless object to slide from one point to another in the least amount of time. It is a problem in the calculus of variations and has important applications in physics and engineering.

5 Must Know Facts For Your Next Test

  1. The brachistochrone curve was first proposed by Johann Bernoulli in 1696 as a challenge to the mathematical community.
  2. The brachistochrone curve is a cycloid, which is the curve generated by a point on the circumference of a rolling circle.
  3. The brachistochrone problem can be solved using the calculus of variations, which deals with finding the functions that optimize a given functional.
  4. The brachistochrone curve has applications in physics, such as the design of roller coasters and other transportation systems.
  5. The brachistochrone problem is an example of a variational principle, where the physical system (in this case, the sliding object) tends to evolve in a way that minimizes a certain quantity (in this case, the time taken to slide down the curve).

Review Questions

  • Explain the key features of the brachistochrone curve and how it is related to parametric equations.
    • The brachistochrone curve is the shape of the path that allows a frictionless object to slide from one point to another in the least amount of time. It is a cycloid, which is a curve generated by a point on the circumference of a rolling circle. The brachistochrone curve can be described using parametric equations, where the x and y coordinates of the curve are expressed as functions of a parameter, such as the angle of the rolling circle. The parametric equations allow for the calculation of the time taken for the object to slide down the curve, which is the key feature that makes the brachistochrone curve an important problem in the calculus of variations.
  • Describe how the brachistochrone problem is an example of a variational principle and how it is solved using the calculus of variations.
    • The brachistochrone problem is an example of a variational principle, where the physical system (the sliding object) tends to evolve in a way that minimizes a certain quantity (the time taken to slide down the curve). The calculus of variations is the branch of mathematics that deals with finding the functions that optimize a given functional, such as the time taken for an object to slide down a curve. To solve the brachistochrone problem using the calculus of variations, one needs to find the shape of the curve that minimizes the time taken for the object to slide from one point to another. This involves setting up an optimization problem and using techniques from the calculus of variations, such as the Euler-Lagrange equation, to find the optimal curve.
  • Analyze the practical applications of the brachistochrone curve in physics and engineering, and discuss how the insights gained from studying this problem can be applied to other real-world problems.
    • The brachistochrone curve has important applications in physics and engineering, particularly in the design of transportation systems such as roller coasters. By understanding the shape of the brachistochrone curve and the principles behind it, engineers can design more efficient and safer transportation systems that minimize the time and energy required for an object to travel between two points. Additionally, the insights gained from studying the brachistochrone problem can be applied to other real-world problems that involve optimizing the path or trajectory of a system, such as the design of spacecraft trajectories, the optimization of manufacturing processes, and the analysis of biological systems. The brachistochrone problem is an excellent example of how the calculus of variations and the study of variational principles can lead to important practical applications and a deeper understanding of the physical world.