In 1696, the Swiss mathematician Johann Bernoulli proposed a challenge to the mathematical community: in the vertical plane, there are countless paths between two points, so which path can make the particle slide from one point to another in the shortest time under the action of gravity, regardless of air resistance and friction?This is the famous descent line problem, which not only demonstrates the practical application of mathematics in the physical world, but is also an important milestone in the development of variational methods.
Mathematical formulation of the problem
The descent line problem can be abstracted into a variational problem, i.e., finding a specific curve among all the smooth curves connecting two points in such a way that the motion time along that curve is minimized. The mathematical model of this problem needs to take into account the length of the path as well as the velocity of the object on the path. By the law of conservation of energy, we know the velocity of an object at a certain height (y), where (g) is the acceleration due to gravity. Thus, the total motion time (t) can be expressed in integral form:
where is the tiny length of the path element.
Applications of the Euler-Lagrangian equation
To find the extremum of this integral, we employ the Euler-Lagrangian equation, which is a powerful tool for solving functional extremum problems for a given boundary condition. By applying the Euler-Lagrangian equation to the integral function, we obtain a second-order nonlinear differential equation with a solution that describes the exact shape of the descent line.
Cycloid: The solution of the most rapid
The solved differential equation shows that the descent line is actually a cycloid. The cycloid is formed by the trajectory of a point on a fixed circle as it rolls along a straight line. Specifically, the parametric equation for the cycloidal is:
x = a(t - sin t)
y = a(1 - cos t)
where (a) is the parameter of the cycloid, and (t) is the parameter variable of the cycloid. This result not only amazed the mathematical elegance of nature, but also inspired later scientists and mathematicians how to describe physical phenomena through the language of mathematics.
The intersection of mathematics and reality
The solution of the descent line problem is not only a triumph of mathematical theory, but also has a profound impact on practical engineering. For example, when designing roller coasters and slides, engineers refer to the principle of the fastest descent line to optimize the passenger experience and safety. In addition, this problem has also promoted the development of variational method, which has a wide range of applications in physics, engineering, economics and many other fields.
Conclusion
The descent line problem is a classic problem that crosses the boundaries of mathematics and physics, which not only provides us with a way to explore the optimal path in nature, but also demonstrates the power of mathematics in explaining and aspiring natural phenomena. Through the study of this problem, we can not only understand the laws of physics more deeply, but also appreciate the infinite possibilities of mathematics in various fields.
References
1] bernoulli, j. (1696). "problema novum ad cujus solutionem mathematici invitantur". acta eruditorum.
2] euler, l. (1744). "methodus inveniendi lineas curvas maximi minimive proprietate gaudentes". lausanne & geneva.