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The hidden secret of Lagrangian mechanics: Why is it simpler than Newton's method?
In the field of physics, Lagrangian mechanics is a classical mechanics statement based on the principle of steady-state action. This method was first proposed by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1760 and further developed in his 1788 book Analytical Mechanics. Lagrangian mechanics describes a mechanical system as a pair (M, L), where M represents the configuration space and L is a smooth function, commonly known as the Lagrangian function.
Lagrange's core concept is energy, not force, which makes the mathematical formulas for dealing with complex systems more abstract and simplified.
Compared to the operability of Newton's laws of motion, Lagrangian mechanics chooses to use energy as its basic component. By giving each object independent generalized coordinates, a new mathematical framework was created that made Lagrange's equations robust to more complex problems. In certain situations, Newton's laws may seem difficult to apply. For example, when calculating the situation in which a ring rolls on a horizontal plane while a bead slides inside it, the varying restraints of the ring and the motion of the bead relative to the ring make it extremely difficult to use Newton's laws.
In general, Newtonian mechanics is suitable for many problems, but in some cases it can be complex and difficult to control. Lagrange's solution offered an elegant shift away from calculating the forces directly and toward calculating the differences in the system's kinetic and potential energies—essentially the sum of the behavior of the dynamical system.
The equations of Lagrangian mechanics allow us to not consider the effects of time-varying constraints, and only the essence of dynamics emerges.
Since Newton's equations of motion usually require three equations for each particle to explain its motion, in the Lagrangian framework, the number of equations is significantly reduced due to the consideration of the reduction of generalized coordinates. In most cases, this shift makes the solution not only effective but also more intuitive. The essence of Lagrangian mechanics lies in its generality, which means that many physical systems can be abstracted and analyzed using a unified mathematical framework.
The role of Lagrangian functions
The Lagrangian function L is defined as the difference between the kinetic energy T and the potential energy V in the system, expressed as L = T - V. This insight guides every trajectory of an object's motion and drives our pursuit of a deeper understanding of physics. In this way, we can summarize the core idea of Lagrangian mechanics in one sentence: the realization of motion is the one with the smallest action among all possible paths.
For many physical systems, simplifying them to point particles and considering their mass and shape can greatly reduce complexity.
In the application of Lagrangian mechanics, when we are in a system composed of N masses m1, m2…, mN, the speed and position of each particle will be accurately described by its position vector. Furthermore, the total kinetic energy of such a system would be the sum of the kinetic energies of the individual massive particles, making the overall energy conversion clearly visible. Unlike Newton's equations, Lagrange used kinetic energy and potential energy to capture the changes in the entire system, reducing the complexity of directly solving for the forces.
From Newtonian Mechanics to Lagrangian Mechanics
Newtonian mechanics is based on the laws of motion, which describe the relationship between mass, acceleration and external forces. For a particle of constant mass, its motion follows Newton's second law of motion F = ma.
However, when the amount of data increases and the system relationships become more complex, Lagrange's equations become more valuable. In the general case, Lagrangian mathematical framework effectively integrates the motion of multiple particles by establishing generalized coordinates for objects, allowing us to stop worrying about the constraints at each moment and focus on the overall behavior.
Therefore, the discovery of Lagrangian mechanics not only changed the way we deal with physical problems, but also allowed us to gain a deeper understanding of the intrinsic connections between things. All this means that in the face of increasing complexity, "energy" provides us with a brand new key to unlock many mysteries that were out of reach in the past.
In future physics research, should we re-evaluate our understanding of basic mechanics and think about the deep connection between Lagrange and Newton?
