Розвиток методики викладання механіки як боротьба варіативного та ньютонівського підходів
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Дата
2020
Назва журналу
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Видавець
СумДПУ імені А. С. Макаренка
Анотація
Чергова стаття з циклу, присвяченому покращення методики викладання механіки. Пропонується новий комплексний підхід викладання механіки, який базується на діалектичному застосуванні багатомірного масиву з декількох базових засад, які комбінуються між собою, у залежності від конкретної ситуації в викладанні, створюючи широке поле можливостей. До рівня «minimum minimorum» масиву засад пропонується включити розгляд розвитку механіки як боротьби та співпраці підходів Ньютона, який використовую узагальнення великої кількості спостережень та дослідів, і варіативних підходів Лагранжа та Гамільтона, які беруть свій початок фактично в оптиці і являють собою вдало вгаданий постулат. Показано, що одночасне використання декількох підходів буде сприяти покращення методики.
A comprehensive approach to teaching mechanics is proposed, which uses a multidimensional array of several basic positions that are combined with each other according to a particular situation. Newton’s mechanics is based on the laws of motion, which are described by differential equations. Fermat offered, at one time, his variant of the extreme principle, which, at the same time, contradicted the principles of Newtonian mechanics. Attempts to build the mechanics on the principle of least action made Euler and Maupertuis, but they cannot be called successful. D’Alembert suggested that the problems of dynamics, in determining the laws of motion, should be reduced to the problems of static, which considers the case of motionless bodies. Lagrange continued development of analytical mechanics in order to reduce mechanics and methods of solving its problems to formulas of general form. It is easy to obtain the d’Alembert principle from the Hamiltonian variational principle. This mechanics makes it quite easy to find a connection with quantum mechanics or relativistic mechanics, which is its great advantage. There are also theories in which the extreme principle of least action can be considered as a consequence of the law of conservation of energy, but not all researchers agree with it. Newtonian mechanics is based on vector concepts such as force vector and momentum vector. Lagrangian mechanics is based on scalar concepts: kinetic energy and Lagrangian power function. Newtonian mechanics must also take into account the forces that support the link. The advantage of Lagrangian mechanics is freedom to choose any coordinate system. The advantage of Lagrangian mechanics is that we are dealing with one general principle of dynamics, the standard procedure of which allows to obtain all the necessary system of equations by using the variational method. In Newtonian mechanics, the movement of the body is from one point in time to another point in time, from one point of space to another. In the mechanics of Lagrange and Hamilton, we see the movement of a separate material point all at once, completely, in four-dimensional space-time. The Newtonian mechanics, when considering motion, considered invariants dynamic quantities – coordinates of material points and their impulses. The dynamic of Hamiltonian mechanics is regarded as a function of some parameters of canonical transformations, behind which are the invariants of a deeper, more fundamental level. Poincare at one time proposed to use as base positions adiabatic invariants, stability theory, differential equation theory, phase space theory.
A comprehensive approach to teaching mechanics is proposed, which uses a multidimensional array of several basic positions that are combined with each other according to a particular situation. Newton’s mechanics is based on the laws of motion, which are described by differential equations. Fermat offered, at one time, his variant of the extreme principle, which, at the same time, contradicted the principles of Newtonian mechanics. Attempts to build the mechanics on the principle of least action made Euler and Maupertuis, but they cannot be called successful. D’Alembert suggested that the problems of dynamics, in determining the laws of motion, should be reduced to the problems of static, which considers the case of motionless bodies. Lagrange continued development of analytical mechanics in order to reduce mechanics and methods of solving its problems to formulas of general form. It is easy to obtain the d’Alembert principle from the Hamiltonian variational principle. This mechanics makes it quite easy to find a connection with quantum mechanics or relativistic mechanics, which is its great advantage. There are also theories in which the extreme principle of least action can be considered as a consequence of the law of conservation of energy, but not all researchers agree with it. Newtonian mechanics is based on vector concepts such as force vector and momentum vector. Lagrangian mechanics is based on scalar concepts: kinetic energy and Lagrangian power function. Newtonian mechanics must also take into account the forces that support the link. The advantage of Lagrangian mechanics is freedom to choose any coordinate system. The advantage of Lagrangian mechanics is that we are dealing with one general principle of dynamics, the standard procedure of which allows to obtain all the necessary system of equations by using the variational method. In Newtonian mechanics, the movement of the body is from one point in time to another point in time, from one point of space to another. In the mechanics of Lagrange and Hamilton, we see the movement of a separate material point all at once, completely, in four-dimensional space-time. The Newtonian mechanics, when considering motion, considered invariants dynamic quantities – coordinates of material points and their impulses. The dynamic of Hamiltonian mechanics is regarded as a function of some parameters of canonical transformations, behind which are the invariants of a deeper, more fundamental level. Poincare at one time proposed to use as base positions adiabatic invariants, stability theory, differential equation theory, phase space theory.
Опис
Ключові слова
методика викладання механіки, варіативний підхід, метод Лагранжа, метод Гамільтона, діалектична взаємодія, парадигма Ньютона, methods of teaching mechanics, variational approach, Lagrange method, Hamilton method, dialectical interaction, Newton’s paradigm
Бібліографічний опис
Корольов, С. В. Розвиток методики викладання механіки як боротьба варіативного та ньютонівського підходів [Текст] / С. В. Корольов // Педагогічні науки: теорія, історія, інноваційні технології : науковий журнал / МОН України, Сумський держ. пед. ун-т ім. А. С. Макаренка ; [редкол.: А. А. Сбруєва, О. А. Біда, Ю. А. Бондаренко та ін.]. – Суми : СумДПУ ім. А. С. Макаренка, 2020. – № 2 (96). – С. 99–110. – DOI: 10.24139/2312-5993/2020.02/099-110.