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Characterization of the maximum principle


People are constantly trying to achieve perfection. In exact mathematical science, this is expressed in the choice of the best solution, even in search of the principles of maximum and minimum.

Influence of the maximum principle

The theory of analytic functions is based on the maximum modulus principle. According to his prescriptions, the maximum modulus of an analytic function, inside any limited area, is always located on the boundary of the specified area.

Thanks to the initial data of this principle, the Schwartz lemma was proved. If we are talking about areas of a specific type, then we can be puzzled by the theorem of Phragmen and Lindelöf and move on to elementary particles, or rather their scattering.

The principle of maximum is affected by the ethics of business communication and, like any of them, gives an employee of a corporation or company basic ethical standards, on which he orients his actions, builds decisions. The principle of maximum progress states that an employee’s behavior and actions will remain ethical as long as they are aimed at achieving results that benefit the organization and its development in terms of morality.

In continuation of this provision, the following statement – the ethical behavior of an employee and an organization can only remain so within the framework of ethical standards established by society.

Maximum Principle – Optimal Control

Maximum problems in optimal control include those that last in time or are long in space. This applies to any constantly changing system, limited by certain limits. Visually, this can be represented in the form of laying the best route through difficult terrain or driving, etc.

Watch the video about the maximum principle in optimal control.

The system must move from the current state to the necessary one with the least expenditure of force and energy and maximum profit or benefit or cohesion, etc. It can be saving anything and getting as a result of any given criterion.

If the optimal solution is already known, then to keep the constant systematic obtaining of the best result can also be applied by applying the theory of optimal control. For example, to stop the oscillation, it is necessary to apply a small counterweight force from different sides until it stops completely. Calculation of the required amount, frequency, and other parameters will lead to its stop.

The theory is equally effective in financial matters, economic, physical processes, etc., and was created by the mathematician Pontryagin with a team of topologists for the economical transfer of a rocket in space between orbits. The gradual narrowing down of possible strategies until the optimal one is identified is the result of that mental attack.

The initial given condition may belong to a linear model with the application of the maximum principle and the only correct solution. But if the original system of conditions is complicated (for example, in the field of space navigation, robotics or quantum systems), then geometric control theory comes to the rescue. In this case, the control possibilities are expanded due to combinations of applied maneuvers different in time and in different order.

Entropy maximum principle

There are two ways to approach the understanding of the principle of maximum entropy:

A striking example of the first interpretation can be recognized as the second law of thermodynamics. The amount of entropy within a particular system in any process remains unchanged or increases, but can never become less. If some system has been isolated and closed for enough time, then the entropy inside it has reached its maximum and an increase is not possible.

This interpretation originates in the theory of probability from the insufficiency of the cause, formulated by Jacob Bernoulli in the 18th century.

Following the second position, the description of the world should take place without personal assessments and prejudices, without hanging various probabilistic outcomes on it. Enumeration should be subject only to what there is no doubt and the reliability is not subject to proof. In this way, possible distortions in the description of the real state of affairs can be avoided.

In the first case, it is not about the system of describing the world, but about the world itself in particular. When the only decision is made how the description of nature will be constructed, at that very moment the choice of nature itself will be determined.

Gottfried Leibniz and his principle of minimum and maximum

The interpretation of the principle of minimum and maximum of the philosopher, mathematician, logician, physicist and thinker in many sciences G. Leibniz is simple. Essence at its minimum allows the development of the maximum of existence. You should not make much effort to achieve the maximum result, since nature usually does everything by itself.

Such a statement is admissible for the reason that the world spreads through processes and phenomena within the sole framework of the law of logic. Everything is simple in it, since the processes proceed continuously, connected, orderly, holistic, that is, perfect.

Obvious examples are:

  • a drop of liquid not in flight, but in space and at rest, without the intervention of any forces from outside. Maximum liquid in the smallest possible surface area;
  • path of light rays in space. The path is always the shortest of most possible. Even refraction cannot change this principle;
  • application of theorems or axioms in solving a large number of problems of the same type. etc.