As we know initially we would be given two points on a 2D surface (consider a plane perpendicular to ground ) we would like to run a small ball from higher level point to lower level point from a track path which we have to make so that it reaches there in minimum time. My idea which i was thinking of: suppose we want to solve it without going into calculus of variations maths stuff and we would like to see what happens to the trajectory in the $v_y$-$v_x$ hodograph. We can conclude some few things regarding the motion from the hodograph, we need the vx component average speed to be maximum so that it can travel that distance in shortest time. And acceleration should be always at its greatest so that between two very close points the time taken would be least. Are the ideas sufficient to tackle for the track path which we want to minimize for the time taken? (My original approach was using calculus of variations which leads to a differential but i wonder if we can use the above idea to solve it )
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$\begingroup$ What have you tried already? $\endgroup$– John AlexiouCommented Feb 23, 2022 at 3:52
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$\begingroup$ @JohnAlexiou Sir the one using calculus of variations $\endgroup$– Orion_PaxCommented Feb 23, 2022 at 5:08
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