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Cleonis | 2 years ago
Interestingly: it is possible to go in all forward steps from Newtonian mechanics to Hamilton's stationary action. That is the approach of this demonstration. (How Hamilton's stationary action came into the physics community is quite a convoluted story. With benefit of hindsight: a transparent exposition is possible.)
Recommended: read the following two articles in this order: Introduction to calculus of variations: http://cleonis.nl/physics/phys256/calculus_variations.php
Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...
The path from F=ma to Hamilton's stationary action goes in two stages: 1) Derivation of the work-energy theorem from F=ma 2) Demonstration that in cases where the work-energy theorem holds good Hamilton's stationary action will hold good also
Also interesting: Within the scope of Hamilton's stationary action there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action.
In the demonstration it is shown for which classes of cases the stationary point corresponds to a minimum of Hamilton's action, and for which classes to a maximum.
The point is: it is not about minimization. The actual criterion is that which both have in common: As you sweep out variation: in the variation space the true trajectory is the one with the property that the derivative of Hamilton's action is zero. The interactive diagrams illustrate why that property holds good (it follows from the work-energy theorem).
Hamilton's stationary action is a mathematical property. When the derivative of the kinetic energy matches the derivative of the potential energy: then the derivative of Hamilton's action is zero.
(Ycombinator does not give control over the layout of the text I submit. I insert end-of-line, to structure the text, but they are eaten.)
dataflow|2 years ago