The Principle of Least Action

Descartes’ Laws of Conservation

Descartes believed that:

• perfectly hard bodies exist
• he discovered the laws of their motion

It started from a reasonable principle: total momentum is conserved in Nature.

Unfortunately, he derived false laws, because his initial principle, the conservation of total momentum, is not true.

The philosophers who followed Descartes noted another conservation law: that of kinetic energy.

It is one-half times the mass times the speed squared.

These philosophers did not derive their laws of motion from this conservation law, but rather the reverse.

They derived the conservation law from their laws of motion.

But kinetic energy is conserved only for perfectly elastic bodies. This reinforced their belief that only elastic bodies exist in Nature.

Momentum is conserved only in special cases.

Kinetic energy is conserved only for certain types of bodies.

Neither one represents a universal principle, a general law of motion.

If one studies these principles and the reasoning their authors used, it is surprising that they ever were able to derive them.

One suspects that they worked backwards from experience, rather than deducing them from basic principles.

Those who argue most rigorously recognize that the principle used to derive the collision laws for elastic bodies would not be applicable to perfectly inelastic bodies.

Moreover, the laws of mechanical equilibrium are independent of the conservation laws used to derive laws for collisions of elastic and inelastic bodies.

The Principle of Least Action

My principle of least action is a principle:

• that is the basis of the laws of motion and equilibrium of all material bodies
• that covers both elastic and inelastic collisions
• so wise and so worthy of the supreme Being
• intrinsic to all natural phenomena.

One observes it at work:

• in every change. and
• in every constancy that Nature exhibits.

In the collision of bodies, motion is distributed such that the quantity of action is as small as possible.

At equilibrium, the bodies are arranged such that, if they were to undergo a small movement, the quantity of action would be smallest.

The laws of motion and equilibrium derived from this principle are exactly those observed in Nature.

This principle applies to all phenomena.

• the movement of animals
• the growth of plants
• the revolutions of the planets