SCIENCE

INVARIANCE IN PHYSICS

Speaker: Victor Suchar on 25 September 1998

Geometrical Principles of Invariance
According to Eugene Wigner, in his Essays, “The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice which permits the complicated nature of the world to be blamed on something which is called accidental and thus permits him to abstract a domain in which simple laws can be found. The complications are called initial conditions; the domain of regularities, laws of nature. Unnatural as such a division of the world structure may appear from a very detached point of view, the underlying abstraction is probably one of the most fruitful the human mind has made. It has made the natural sciences possible.”
The possibility of abstracting laws of motion from the chaotic set of events that surround us is based on two circumstances. First, in many cases a set of initial conditions can be isolated, which is not too large and yet it contains all the relevant conditions for the events under consideration. In the classic example of the falling body, one can disregard almost everything except the initial position and velocity of the falling body - its behaviour will be the same and independent of the degree of illumination, the neighbourhood of other objects, their temperature, etc. Second, it is also essential, that given the same initial conditions, the result will be the same no matter when and where we realise them. This principle can be formulated as the statement that absolute position and absolute time are never essential initial conditions, and it constitutes the most important theorem of invariance in physics. The geometric theories of invariance in physics, postulate in addition, the irrelevance to initial conditions of orientation and of the state of motion, as long as it remains uniform, free of rotation and on a straight line.

Variational Principles
There is another type of theory of invariance, fundamental in physics, based
on the Principle of Least Action, a variational principle which states that
“AMONG ALL POSSIBLE MOTIONS, NATURE REACHES ITS GOAL WITH MINIMUM EXPENDITURE OF ACTION”.
While in Newton’s mechanics the action of a force is measured by the momentum produced by the force, Leibniz advocates another quantity - “Vis Viva” (living force) as the proper gauge for the dynamical action of a force. Vis Viva coincides, apart from the factor of 2, with the quantity of motion called today “kinetic energy”. Leibniz replaced Newton’s “momentum” with “kinetic energy” and “force” by the “work function”. Leibniz is thus the originator of the second branch of mechanics usually called “analytical mechanics” which bases the entire study of equilibrium and motion on two fundamental scalar (magnitude but no direction) quantities: “kinetic energy” and “work function”, later replaced by “potential energy”. Since motion is by its very nature a directed phenomenon, it seems puzzling that two scalar quantities should be sufficient to determine it. The “Energy Theorem” which states that the sum of kinetic and potential energies remains unchanged during the motion, yields one equation, while the motion of a single particle in space requires 3 equations (in the case of a mechanical system of two or more particles the discrepancy becomes even greater). And yet, it is a fact that these two fundamental scalars contain the complete dynamics of even the most complicated material system, provided that they are used as the basis of a principle, the principle of least action, rather than of an equation.
This was the basis of the Lagrange and Hamilton’s formulations, which use the principle of least action as an organising principle in order to express all laws of Newtonian physics as a representation of minimum problems. (The
speaker described in some detail their development.)
The Hamiltonian of a system is the energy expressed in terms of position and momentum of a particle. Given the Hamiltonian, equations can be written and solved, which give the orbits of the particles in terms of a set of initial conditions. This is the most complete expression of determinism in classical mechanics, and it is vital to the theories of magnetism and relativity and to the development of quantum mechanics. At its heart, this expression has the teleological concept of minimal action, called by some (e.g. Mach) as being of essentially an “economic character” and by others as “metaphysical”.
The calculus of variations was thus the basis for a theory of invariance derived from a theory of motion. In modern physics, the Principle of Relativity requires that the laws of nature shall be formulated in an invariant fashion, i.e. independently of any frames of reference. The methods of the calculus of variations automatically satisfy this principle, because the minimum of a scalar quantity does not depend on coordinates. While the Newtonian equations of motion did not satisfy the principle of relativity, the principle of least action remained valid. This principle has been transmitted into quantum mechanics through Bohr's correspondence principle and the use of the Hamiltonian in Dirac's formulations.
The Principle of Least Action, a metaphysical principle, stands as basic to both fundamental theories in physics - the formalisms of classical and quantum mechanics.
Victor Suchar