The river flows from outskirt of the mountain downward following a very crooked path. The lava from the volcanic activities chooses a similar route. Similarly, a glacier pathway incudes a number of curvatures. Furthermore, The one who is skiing tries to control himself against gravity by following a zigzag path. I conclude that nature is quite nonlinear. What is your opinion? Can we simulate this crooked trajectory by a set of differential equations? I do not think it would be so easy!
Moreover, the engineers use the curvatures to build the mountainous roads and even looking at the railroad tracks, most of them do not follow a straight line.
Very interesting question. It seems to be a randomness in the formation of river channels.
Zibret and Verbov tried to simulate randomness observed in the geometry of major river channels Amazon, Mississippi, Danube and Sava using Serpinski triangle and fractal models :
"Quantitative analysis of randomness exhibited by river channels using chaos game technique: Mississippi, Amazon, Sava and Danube case studies"
Nonlinear Processes in Geophysics, Volume 16, Issue 3, 2009, pp.419-429
http://adsabs.harvard.edu/abs/2009NPGeo..16..419A
But they could not specify a general model to generate the river path-ways. They concluded that "it is crucial to study the results for several different rivers in different climatic and geological settings before drawing any final conclusions"!
I may resolve that the natural routes and trajectories resemble to be croocked or curved, otherwise in order to cover a straight line we may consume much more energy than a crooked one!?
As a consequence, there is likely no straight pathway in nature.
A ballistic projectile trajectory in air is not a parabola but a more complex curvature:
Take a look on grand tour of Voyager1 and 2 in our solar system;The tracks are two large curvatures in space:
Wikipedia:
If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation.
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