Concept explainers
Problems
Falling Bodies. Let
The action integral and Euler-Lagrange equation for the height above the Earth’s surface of a body of mass
Answer to Problem 1P
Solution:
The action integral for
The Euler-Lagrange equation for
Explanation of Solution
Given information:
Let
Explanation:
The action of the integral of a system is defined by
The kinetic energy of a falling body above the earth’s surface subject only to Earth’s gravitational acceleration is,
By using velocity as the derivative of height,
The potential energy of a falling body above the earth’s surface subject to the gravitational acceleration is,
Thus the action integral is
The Euler-Lagrange equation for the functional
From the action integral
Hence
Consider
Therefore, the Euler-Lagrange equation for
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