I’m confused about the motivation for L=α+V in terms of time dilation in general relativity. I was under the impression that general relativity doesn’t even have a notion of gravitational potential, so I’m not sure what this would mean. And in Newtonian physics, potential energy is only defined up to an added constant. For α+V to represent any sort of ratio (including proper time/coordinate time), V would have to be well-defined, not just up to an arbitrary added constant.
I also had trouble figuring out the relationship between the Euler-Lagrange equation and extremizing S. The Euler-Lagrange equation looks to me like just a kind of funny way of stating Newton’s second law of motion, and I don’t see why it should be equivalent to extremizing action. Perhaps this would be obvious if I knew some calculus of variations?
Well, it makes sense for the effective field theory form of GR, for light at least.
The key to remembering how to derive the Euler-Lagrange equation (for me) is to remember that the variation in L vanishes at the boundary. This is what’s going to let you do an integration by parts and throw away the constant term. Actually, once you have an intuitive grasp of what’s going on, it’s kind of fun to derive generalized EL equations for Lagrangians with more complicated stuff in them.
I’m confused about the motivation for L=α+V in terms of time dilation in general relativity. I was under the impression that general relativity doesn’t even have a notion of gravitational potential, so I’m not sure what this would mean. And in Newtonian physics, potential energy is only defined up to an added constant. For α+V to represent any sort of ratio (including proper time/coordinate time), V would have to be well-defined, not just up to an arbitrary added constant.
I also had trouble figuring out the relationship between the Euler-Lagrange equation and extremizing S. The Euler-Lagrange equation looks to me like just a kind of funny way of stating Newton’s second law of motion, and I don’t see why it should be equivalent to extremizing action. Perhaps this would be obvious if I knew some calculus of variations?
Well, it makes sense for the effective field theory form of GR, for light at least.
The key to remembering how to derive the Euler-Lagrange equation (for me) is to remember that the variation in L vanishes at the boundary. This is what’s going to let you do an integration by parts and throw away the constant term. Actually, once you have an intuitive grasp of what’s going on, it’s kind of fun to derive generalized EL equations for Lagrangians with more complicated stuff in them.