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The Secret of Lagrangian and Hamilton: How to Find Constant from Symmetry?
In the field of physics, there are some key concepts that can help us gain a deeper understanding of the nature of motion. Among them, the constants of motion - such as energy, momentum, angular momentum, etc. - play an important role in describing the behavior of the system. These constants are not only basic conserved quantities in mechanical motion, but also key factors in outlining the dynamics of physical systems.
The movement of constants is not only a mathematical constraint, but also a fundamental reflection of the system's behavior.
In classical mechanics, a constant of motion is a physical quantity that remains unchanged over time. These quantities are not affected by external influences and can therefore provide deep insights into the behavior of a system without fully solving the equations of motion. For example, through Poinsot's construction, we can see that the motion trajectory of a rigid body rotating without torque is the trajectory formed by the intersection of the conservation of total angular momentum and the conservation of energy. This is quite complicated in mathematics, but quite intuitive in application.
So, how can we determine these constants? There are several ways to identify constants of motion, the simplest of which often rely on intuitive assumptions. In this process, researchers may assume that a certain quantity is constant based on experimental data, and then use mathematical methods to prove that the quantity is indeed conserved during motion.
The movement of constants not only reflects the symmetry of physical systems, but also the profound abstraction behind the laws of physics.
Another common approach is to find the constants of motion via the Hamilton–Jacobi equations, which works particularly well when the Hamiltonian takes an easily recognizable functional form. However, Lagrange's symmetry principle has a different meaning. It points out that if the Lagrangian quantity remains unchanged under a certain transformation, then the quantity corresponding to this transformation is a conserved quantity. According to Noether's theorem, the conservation of energy comes from the Lagrangian invariance to time translation, while the conservation of momentum comes from the invariance to space translation, and the same is true for rotation.
The identification of these conserved quantities allows us to simplify physical systems into more understandable models. In ongoing research, scholars have found that if the Hamiltonian of a system commutes with a certain quantity and this quantity does not obviously depend on time, then this quantity is a constant of motion, which provides strong support for the observation quantity in quantum mechanics.
However, things get complicated when we get into quantum mechanics. At this point, we often find that energy is no longer the only conserved quantity. Every observable in phase space can become a constant of motion if it commutes with the Hamiltonian. This provides us with another perspective to understand the behavior of quantum systems.
The conservation of observable quantities provides a theoretical basis for the stability of quantum systems.
It is worth noting that the study of quantum chaos shows that for a non-integrable system, energy is the only conserved quantity. This points to a fundamental difference between quantum chaotic systems and integrable systems: in integrable systems we can find multiple constants, while in non-integrable systems we only get one constant, energy.
The existence of these constants has profound implications for the entire theory of physics, both in classical mechanics and quantum mechanics, and emphasizes the importance of structure and symmetry in understanding the universe. The identification and understanding of each conserved quantity not only brings a fundamental understanding of motion, but also promotes the development of physics theory.
Overall, both the process of finding constants of motion through symmetry and the complex interrelationships between a range of theories are fascinating ways to explore the physical world. But in the process, perhaps we should think about how these symmetries and conservation quantities guide humans to a deeper understanding and exploration of the universe?
