МЕТОДИЧНЕ ОБҐРУНТУВАННЯ ВАРІАЦІЙНОГО ПРИНЦИПУ ГАМІЛЬТОНА-ОСТРОГРАДСЬКОГО

Abstract

It is stated on the basis of the current research analysis that the foundation of the bases of analytical methods of research of physical systems and variation methods have remained outside the attention of methodological science and, therefore, coverage of methodological aspects of foundation the topic “Variation integral principle of stationary action” in teaching theoretical physics in pedagogical universities is relevant.Based on the fact that the mechanical state at each moment is clearly defined as a configuration system as well as velocities system, we introduce some arbitrary dynamic function of the system state that is connected to the coordinates and velocities of all material points and perhaps time. The value of this function at each point in time, we multiply by dt and integrate for a period of time : If we distract from the action of specified forces, and consider only the effect of superimposed ties, it is clear that without breaking the connection with the bodies which limit the movement of material points, the system from position 1 to position 2 during the same period of time could move a large number of close routes, which are not described by equations of motion, and answer only to the linkage equation. To set the criterion that distinguishes the real (direct) path of the system from all other kinematically possible (virtual) for given links, we determine the variation of the resulting integral and come to the conclusion that the actual motion of the mechanical system between two given positions q(t1) and q(t2) configuration space occurs along the trajectory for which the action functional S(q, t1, t2) acquires a stationary value, and, in addition, we obtain the Lagrange equation which, like Newton’s second law, is equations of motion. To consolidate the material, students are invited to consider the problem of the movement of the harmonic oscillator and determine the magnitude of the action between two points along the trajectory and along another close line between the same points. This technique allows students to form a sufficiently deep and stable understanding of this principle and gives an idea of the use of this principle to solve problems of mechanics, which are difficult to solve using only Newton’s laws. Key words: methods of teaching physics, variation principles, Lagrange equations, integral of action, future teachers of physics.