Specifically about the Lagrangian of Classical Mechanics (Hamilton's Action) I have discussed that on physics.stackexchange https://physics.stackexchange.com/a/

Specifically about the Lagrangian of Classical Mechanics (Hamilton's Action) I have discussed that on physics.stackexchange https://physics.stackexchange.com/a/670705/ https://physics.stackexchange.com/a/670705/ The ideas are expressed in diagrams. What is expressed in the diagrams is repeated/corroborated in mathematical expressions (stackexchange supports Mathjax). The visualizations leave no room to get lost.