Lagrangian and Hamiltonian mechanics are two mathematical formulations used to describe the motion of particles and systems in classical mechanics. They are based on the principle of least action and provide a more systematic and elegant approach compared to Newton's laws of motion.
No, you do not need to know calculus of variations to understand Lagrangian and Hamiltonian mechanics. While calculus of variations is the mathematical foundation of these formulations, it is not necessary to have a deep understanding of it to grasp the concepts of Lagrangian and Hamiltonian mechanics.
One of the main advantages of using Lagrangian and Hamiltonian mechanics is that they provide a more systematic and elegant way to describe the motion of particles and systems compared to Newton's laws of motion. They also allow for a more general and unified approach, making it easier to apply to different physical systems.
Yes, Lagrangian and Hamiltonian mechanics can be applied to quantum mechanics. In fact, they are the basis for the formulation of quantum mechanics known as quantum field theory. However, in quantum mechanics, the Lagrangian and Hamiltonian are expressed in terms of quantum operators instead of classical variables.
Lagrangian and Hamiltonian mechanics have various real-life applications, including analyzing the motion of celestial bodies, predicting the behavior of mechanical systems such as pendulums and springs, and understanding the dynamics of particles in quantum field theory. They are also used in engineering and physics to design and optimize systems and processes.