solveforX said:Hey, in general relativity, essentially I am asking how any metric (I.e. schwarzschild metric) was found. are the metrics derived or are they extrapolated from the correct lagrange equations of motion? If there is a derivation available, please provide a link.
thanks
The Lagrangian is a mathematical function that describes the dynamics of a physical system. It is derived from the principle of least action, which states that the path of a system between two points in time is the one that minimizes the action, a quantity related to the energy of the system.
The equations of motion can be derived from the Lagrangian using the Euler-Lagrange equations, which relate the partial derivatives of the Lagrangian with respect to the system's variables to the forces acting on the system. By solving these equations, one can determine the trajectory of the system over time.
Yes, the Lagrangian approach can be used to describe the dynamics of any physical system, from simple objects moving in a straight line to complex systems such as planets orbiting around the sun. However, the Lagrangian may be more useful for certain types of systems, such as those with constraints or varying forces.
Yes, the Lagrangian can be extended to include non-conservative forces, such as friction or air resistance. This is done by adding additional terms to the Lagrangian that account for these forces, allowing for a more accurate description of the system's dynamics.
The Lagrangian approach is a more general and elegant way to describe the dynamics of a system compared to Newton's laws of motion. While Newton's laws focus on the forces acting on a system, the Lagrangian takes into account the entire path of the system and minimizes the action. This approach also allows for more complex systems to be described, as it does not rely on specific force laws.