ผมก็ไม่เคย derive เรื่องนี้
ไว้ผมจะทำหลังจากผมตอบว่าก้อนหิน"รู้ว่า"จะตกอย่างไรได้ก่อน (ก้อนหินถามกฏธรรมชาติที่ไหนถึงรู้ว่าต้องเปลี่ยนความเร็วอย่างไร)
ลองดูประวัติเรื่องนี้เสียหน่อย:
http://www.mathpages.com/rr/s3-04/3-04.htm"It's worth noting that although Fermat was closer to the truth, his principle of "least time" is not strictly correct, because the modern formulation of "Fermat's Principle" states that light travels along a path for which the time is stationary, (i.e., such that slight transverse changes in the path don't affect its length), not necessarily minimal. In fact, it may even be maximal, as can be verified by looking at yourself in the concave surface of a shiny spoon. The "reason" that light prefers stationary paths can be found in the theory of quantum electrodynamics and Feynman's "sum over all paths" interpretation, which shows that if neighboring paths take different amounts of time, the neighboring rays arrive at the destination out of phase, and cancel each other out, whereas they reinforce each other if the neighboring paths take the same amount of time, or differ by some whole multiple of the wave. A stark demonstration of this is given by diffraction gratings, in which the canceling regions of a mirror are scraped away, resulting in reflective properties that violate Hero's law of equal angles. "
และ
http://scitation.aip.org/journals/doc/AJPIAS-ft/vol_71/iss_5/423_1-F1.jpg(จาก
http://www.eftaylor.com/pub/call_action.html)
และ
http://www.eftaylor.com/leastaction.html
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