Principle of Least Action

Le grand principe que la nature, dans la production de ses effets, agit toujours par les voies plus simples (Maupertuis, Œuvres, 1752).

Pierre-Louis Moreau de Maupertuis in these words introduced a mechanical rule of breathtaking simplicity and considerable reach.  This law of nature is known as the “Principle of Least Action” and it orchestrates every movement in the Universe down to the behaviour of the smallest fragments of the strange quantum world[1].

More specifically, this principle states that the universe unfolds its events in order to make its Kinetic Energy MINUS its Potential Energy a minimum over the time taken for an event to occur.

This is not a principle that lends itself immediately for adoption by the human intuition, but it can be exemplified readily.  In the previous section we examined the energy transformations that occur through the flight of a bouncing ball.  We should now return to this ball to review its trajectory through to the first bounce.

First Bounce of Ball

A thrown ball has a single preferred trajectory from many possible alternatives

There are many paths that the ball might take from launch to its first bounce.  Three simple paths are drawn in figure (1) as A, B and C, but there are many other logically possible twisted or spiralling loci a ball might take to return to the ground.  However, for a particular throw, there is only one trajectory allowed by the Principle of Least Action.  This principle determines that Kinetic Energy minus Potential Energy (KE-PE) throughout the flight must be lowest of all possible values.[2]

Imagine what would happen if the projectile drawn above would turn out to be an egg rather than a bouncing ball.  The trajectory through to the first bounce would remain unchanged, and then as the egg hits the ground all the energy would be momentarily transformed into strain energy within the eggshell.  As with all things, the shell has a limited capacity to store this strain energy and this capacity would be exceeded.  Excess energy would then be converted into the surface energy of the various fragments of a shattered eggshell, and there might well be residual kinetic energies left to spread the ensuing debris over a wide area.  The breaking of the egg and the potential energy in the newly created surfaces, their surface energy, are also determined by the Principle of Least Action.

The origins of the Principle of Least Action can be traced back to Aristotle in ancient musings on the fundamental economy in nature.  Maupertuis pondered that the perfection of the Universe demanded an Aristotelian economy of nature that is opposed to a needless expenditure of energy, thus sketching out the Principle.  The idea was passed from Maupertuis to the Swiss mathematician Leonhard Euler and then onto the Italian Joseph Louis Lagrange.  The “Lagrangian” became a mathematical description of the Principle of Least Action that forms a foundation stone for the field of theoretical mechanics.  Half a century later, William Rowan Hamilton forged an equivalent mathematical analysis based on what is now called the “Hamiltonian” providing a description of the total mechanical energy of a system.

Energy was thus adopted in the 18th and 19th centuries as an underpinning concept and rational explanation of physical observations of the natural behaviour of the world and universe.  And at the very centre of this grand unification was the Principle of Least Action.  Newton’s Laws of Motion and the Conservation of Energy are consequences of this Principle.  James Clerk Maxwell extended the principle to apply to electrical phenomena, so that forms of electromagnetic energy could also be encompassed.  The formula has endured even in the 20th century where the Lagrangian function remains a cornerstone as Richard Feynman, the American physicist, proposed new formulae for the quantum world based upon the Principle of Least Action[3].

The stochastic failure of the Viscoelastic Sale Event i analogous to the dissipation of energy in a viscoelastic system, which is itself a consequence of the Principle of Least Action operating in this specific domain.  In fact, this stochastic failure has much in common with the breaking of an egg described above, where the fractures occur on a microscopic scale.

Whether it is by metaphor or mechanism, this Viscoelastic Sale Event is proposed as a means to expand the Principle of Least Action to apply to the creation of value through innovation.

 

Notes:

[1] An excellent resource to understand the Principle of Least Action can be found on: www.eftaylor.com/leastaction.html

[2] The potential energy increases with height between A and B, and so KE-PE is reduced making B the preferred trajectory.  This works up to a point.  However, for still higher trajectories a higher initial kinetic energy is required, so that the value of KE-PE will start to rise despite the further reduction of the potential energy.  The Principle of Least Action will thus not allow these higher trajectories, making B the special path of least action that is actually traced by the projectile.

[3] Richard Feynman’s doctoral thesis was entitled “The Principle of Least Action in Quantum Mechanics”,  Princeton University, dated 4th May 1942.

 

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