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From planetary motion to electron hopping: How is Lagrangian mechanics changing science?
In the world of physics, the emergence of Lagrangian mechanics is like a breath of fresh air, gradually rewriting our understanding of motion. Since the Italian-French mathematician and astronomer Joseph-Louis Lagrange published this theory in the 18th century, the scientific community's view of motion has undergone unprecedented changes. Lagrangian mechanics is based on the principle of state dose (i.e. the principle of minimum action) and uses the concept of energy rather than force to describe mechanical systems, allowing many complex problems to be easily solved.
By establishing the independent generalized coordinates of all objects, Lagrangian makes the movement of the mechanical system not the effect of a single force, but the balance of the energy of the entire system.
In traditional Newtonian mechanics, the analysis of many systems has become extremely complicated due to excessive focus on the role of external forces. For example, imagine a ring rolling on a horizontal surface with a pearl sliding inside it. Such a system involves time-varying constraining forces, making using Newton's equations to calculate the motion of the ring almost a nightmare.
However, Lagrange succeeded in describing these systems in an original way. He calculated behavior by summing up all possible paths of motion, thus arriving at the so-called "action." The extreme values of this quantity correspond to the path of motion actually traveled by the particle, omitting an explicit calculation of the binding force.
The core of Lagrangian mechanics lies in the Lagrangian function, which is a summary function of energy that simplifies the dynamics of the entire system.
According to Lagrangian theory, the Lagrangian function can be expressed as the difference between kinetic energy and potential energy, that is, L = T - V. This approach allows the dynamic behavior to be described by Lagrangian equations without considering constraint forces. In addition, Lagrangian mechanics also allows a system of N point particles to be described by simplified coordinates, thus greatly reducing the number of equations that need to be solved.
When we compare Lagrangian mechanics with Newtonian mechanics, it is not difficult to find that this theory is not just a linguistic conversion. The use of Lagrangian functions can be regarded as a reconstruction in physics, allowing the behavior of the system to be accurately predicted under changing constraints.
In Lagrangian mechanics, for each particle of the system, its independent coordinates and speed can be defined, which makes dynamic expression a more common form.
As time goes by, Lagrangian mechanics is not only applied to problems in classical physics, but also begins to quietly enter the field of quantum mechanics. Lagrangian principles were used to derive the actions in quantum field theory, further cementing its importance in physics. This shows that Lagrangian mechanics is not only a tool to describe motion, but also an analytical framework suitable for a variety of physical phenomena.
Today, Lagrange's ideas still shine in every corner of scientific research. From the design of aircraft to the behavior of microscopic particles, Lagrangian principles provide us with a new perspective on understanding the physical world. However, we can't help but ask, will this new way of describing lead us to a deeper level of understanding of physics?
