The left side seems ok. I don't know where the right side comes from, I mean the second factor of the left side.dynamicskillingme said:
Point transformations in the Lagrangian refer to changes in the coordinates used to describe a physical system. These transformations can help simplify the equations of motion and reveal underlying symmetries in the system.
Point transformations are important in Lagrangian mechanics because they allow us to find new sets of coordinates that make the equations of motion easier to solve. They also help us identify conserved quantities, such as energy and momentum, which are crucial in understanding the behavior of a system.
Point transformations affect the Lagrangian by changing the coordinates used to describe the system. This results in a new Lagrangian that may have a simpler form or reveal new symmetries. The equations of motion derived from this new Lagrangian will be equivalent to the original equations of motion, but may be easier to solve.
Some common examples of point transformations in the Lagrangian include translations, rotations, and scale transformations. These transformations can be applied to the coordinates of a system to simplify the equations of motion and reveal symmetries.
Point transformations are closely related to Noether's theorem, which states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. Point transformations can help identify these symmetries and the associated conserved quantities, providing a powerful tool for understanding the behavior of a system.