(صورة توضيحية فقط)
A reformulation of classical الميكانيكا based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy ([latex]L = T – V[/latex]). The equations of motion are derived from the Euler-Lagrange equation, [latex]\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) – \frac{\partial L}{\partial q_i} = 0[/latex], using generalized coordinates, which simplifies analysis of complex systems with constraints.
تُقدم ميكانيكا لاغرانج، التي طورها جوزيف-لويس لاغرانج، بديلاً قوياً وأنيقاً لصياغة نيوتن. فبدلاً من التركيز على القوى والتسارعات، وهي كميات متجهة، تُركز على الطاقات، وهي كميات قياسية. هذا التحول في المنظور غالباً ما يُبسط المسائل بشكل كبير، وخاصةً تلك التي تتضمن قيوداً.
The central concept is the principle of stationary action. It posits that the path taken by a physical system between two points in time is the one for which the ‘action’ is stationary (a minimum, maximum, or saddle point). The action is defined as the time integral of the Lagrangian function, [latex]S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt[/latex]. The Lagrangian, [latex]L[/latex], is defined as the kinetic energy [latex]T[/latex] minus the potential energy [latex]V[/latex] of the system.
By applying the calculus of variations to find the path that makes the action stationary, one derives the Euler-Lagrange equations. A key advantage of this approach is the use of generalized coordinates ([latex]q_i[/latex]). These are any set of parameters that uniquely define the configuration of the system. For example, for a double pendulum, the two angles are natural generalized coordinates. This freedom to choose the most convenient coordinate system is a major strength. Furthermore, forces of constraint (like the tension in a pendulum rod) do not appear in the Lagrangian formulation, as they do no work, meaning they can be ignored, greatly simplifying the equations of motion for constrained systems.
This formalism is not only a powerful tool in classical mechanics but also serves as the foundation for more advanced theories, including quantum mechanics (through Feynman’s path integral formulation) and quantum field theory.
ميكانيكا لاغرانج
(إذا كان التاريخ غير معروف أو غير ذي صلة، على سبيل المثال "ميكانيكا الموائع"، يتم توفير تقدير تقريبي لظهوره الملحوظ)
الصور والتنزيلات بالحجم الكامل متاحة فقط، ومجانية تمامًا، للأعضاء المسجلين.
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