Lagrangian Mechanics is a mathematical framework used to describe the motion of a system of particles under the influence of forces and constraints. It is based on the principle of least action, which states that the actual path of a system between two points in time is the one that minimizes the total action of the system.
Constrained systems in Lagrangian Mechanics refer to systems in which the particles are subject to constraints such as fixed relationships between their positions, velocities, or accelerations. These constraints must be taken into account when applying the principle of least action to determine the equations of motion for the system.
While Newtonian Mechanics is based on the concept of forces and accelerations, Lagrangian Mechanics is based on the principle of least action. This allows for a more elegant and powerful approach to solving problems in mechanics, as it does not require the concept of forces and can handle more complex systems with constraints.
One of the main advantages of using Lagrangian Mechanics is its ability to handle complex systems with constraints, where Newtonian Mechanics may become cumbersome. It also provides a more intuitive and geometric understanding of the motion of a system, as well as allowing for easier application of variational methods.
Lagrangian Mechanics is applied in a wide range of real-world situations, from classical mechanics problems in physics to more complex systems in engineering and biology. It is particularly useful in fields such as celestial mechanics, fluid dynamics, and robotics, where constraints play a significant role in the behavior of the system.