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Swinburne University of Technology Sarawak Campus

The principle of least action

October 3, 2012

By Dr Manas Kumar Haldar

The readers of this article who are not physicists might find the title bizarre. But believe me, the principle of least action is quite old and is a corner-stone of physics.

Sir Issac Newton was the first person to enunciate the laws of motion. His second law states that the force on a particle is equal to its mass times its acceleration. Knowledge of the force and the mass allows one to calculate the path (trajectory) of a particle. However the formulation gets quite complicated when applied to a system with many possible movements. A more elegant formulation of mechanics was propounded by the French physicist, Joseph Louis Lagrange in 1788.

In this formulation, an expression called Lagrangian is formed in terms of the kinetic and potential energies of the system.

Kinetic energy is the energy due to motion. Potential energy is the energy due to the position. For example, a body raised above the ground has potential energy due to the gravitational pull of the earth. These energies and hence the Lagrangian can be written in terms of the velocity and position of different parts of a system.

The integral of the Lagrangian is called action. A reader unfamiliar with calculus might feel uncomfortable. But let us accept it as a definition. The principle of least action states that the paths followed by the different parts of a system are those that minimize the action. The minimum of action gives rise to a set of equations of motion for the system.

“Fine”, one might say, “but is there any practical application for this principle?” Yes indeed, one applies this principle in robotics. A robot as you know has many parts, for example, arms, grips etc. These parts can perform various types of motion such as rotation and translation. The principle of least action allows us to obtain the equations of motion of these parts in a systematic way.

The principle of least action is closely related to another minimizing principle of physics called the principle of least time. This was formulated by the French amateur physicist and mathematician, Pierre de Fermat. The principle relates to the path of light rays rather than path of a particle. It states that the path taken by a ray of light travelling from one point to another is such that the time of travel is a minimum. This is why light travels along a straight line in air. A straight line is the shortest path between two points in air which makes the time of travel between the two points in air minimum. Other phenomena, such as the laws of reflection and refraction of light, can also be obtained from this principle.

The general reader might be getting bored with all the physics and mathematics at this point. So in a lighter vein, one can ask if the principle of least action is of any use in human life. Unfortunately, neither action nor minimum in human endeavour is as precisely defined as in physics. It is a grey area. Let me venture away from my specialization and consider economics.

In economics, there is a principle called “Laissez-faire” which essentially means “let the government leave the market alone”. This was advocated by the famous economist, Adam Smith, although he did not use the phrase “Laissez-faire” (It was coined by French finance minister Jean Baptiste Colbert in 1680).

Adam Smith did not suggest total lack of government control. In fact, the “market” is mean and “short sighted”. Left to itself, it does not consider many important human issues such as welfare and the environment. So the government must exercise some control. Adam Smith and his followers advocate minimal control of markets. In other words, the proper economic “path” is that for which the government follows the principle of least action on markets. History shows that economies with less control have been far more successful than those under complete government control, such as those in some communist countries.

What about politics? Least action can lead to tragedies as we have seen in the events in Srebrenica in Bosnia. On the other hand, politicians in America, China and Russia suggest that a drastic action, such as the bombing of Iran, can cause major turmoil in the Middle East. Negotiations and economic pressures are preferred. Let us see if this principle of least action works out to the benefit of all.

An issue that may be badly affected by least action is global warming. It has been scientifically established that activities of human beings are causing this warming. If unchecked, it can give rise to severe consequences, such as flooding, severe storms, desertification and spread of disease. Governments are dragging their feet on this issue. Is this least action beneficial?

Dr Manas Kumar Haldar is an associate professor with the Faculty of Engineering, Computing and Science. He is contactable at mhaldar@swinburne.edu.my