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Although not the same, I remember the first time I encountered the description of dynamic systems using Lagrangian mechanics and it was beautiful. It finally ma

Although not the same, I remember the first time I encountered the description of dynamic systems using Lagrangian mechanics and it was beautiful. It finally made sense. Instead of doing what seemed like some mumbo-jumbo sarting from a static system, that, eventually and through convoluted ways, lead to the equations we were kind of looking for, here was a clean and logical way to look at what was relevant to the dynamics of the system, with an infaillible, straightforward, mathematical way of getting the equations of motion. I'm not that familiar with Hamiltonian formulations, but its conservation properties could bring some important improvements to the current way the Navier-Stokes equations are treated. Conservation of linear and angular momentum, for a start, could be nice.. Now, let's see if I understand anything from this paper..

deleted 3y ago
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om2 3y ago
I have tried to teach myself both Hamiltonian and Lagrangian classical mechanics and there’s one mental hurdle I have not been able to get over. The problems are generally set up with starting position and momentum known, and ending position and momentum known, and then the math tells you the path taken along the way. But what if the ending position and momentum is unknown? How does one use these formulations of mechanics to predict the future and not just postdict the past? Is this just how beginner problems tend to be set up?
- joshjob42 3y ago
  Hm, well typically in Lagrangian formulations, you're right that you define a functional called an "action", and this involves some integral from an initial to a final position in configuration space. Then the idea is that the system will evolve in such a way as to minimize this action. A stationary point (like a maximum or minimum) has gradient 0. So a condition for this "principle of least action" to be realized is that the "gradient" (called the variation here) of the action is 0 for the real path. That is, if you take a path the system carves out through the configuration space and perturb that path, and you compute the total effect on the action from that small perturbation, you find it should be 0. You do this by actually taking this derivative and you find that you can guarantee that the differential of the action is 0 if the system takes a path which is the solution to a set of differential equations, and you can generally find the solution to those differential equations only with information about the origin, ie you don't need both the start and end conditions to find a unique solution. So you're right, it's a bit weird conceptually. You sort of start saying "the system obeys a path that minimizes the action between it's initial and final positions" and then find that this produces a set of conditions which form a system of diff eqs that you can find general solutions for and select out a unique solution just with the initial conditions, no need for the final condition.
- fooker 3y ago
  Good question. The answer is, unfortunately, boring. You solve equations to find the unknowns. If there are too many unknowns there are too many solutions and the whole thing is useless.

deleted 3y ago

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om2 3y ago

I have tried to teach myself both Hamiltonian and Lagrangian classical mechanics and there’s one mental hurdle I have not been able to get over. The problems are generally set up with starting position and momentum known, and ending position and momentum known, and then the math tells you the path taken along the way. But what if the ending position and momentum is unknown? How does one use these formulations of mechanics to predict the future and not just postdict the past? Is this just how beginner problems tend to be set up?

joshjob42 3y ago
Hm, well typically in Lagrangian formulations, you're right that you define a functional called an "action", and this involves some integral from an initial to a final position in configuration space. Then the idea is that the system will evolve in such a way as to minimize this action. A stationary point (like a maximum or minimum) has gradient 0. So a condition for this "principle of least action" to be realized is that the "gradient" (called the variation here) of the action is 0 for the real path. That is, if you take a path the system carves out through the configuration space and perturb that path, and you compute the total effect on the action from that small perturbation, you find it should be 0. You do this by actually taking this derivative and you find that you can guarantee that the differential of the action is 0 if the system takes a path which is the solution to a set of differential equations, and you can generally find the solution to those differential equations only with information about the origin, ie you don't need both the start and end conditions to find a unique solution. So you're right, it's a bit weird conceptually. You sort of start saying "the system obeys a path that minimizes the action between it's initial and final positions" and then find that this produces a set of conditions which form a system of diff eqs that you can find general solutions for and select out a unique solution just with the initial conditions, no need for the final condition.
fooker 3y ago
Good question. The answer is, unfortunately, boring. You solve equations to find the unknowns. If there are too many unknowns there are too many solutions and the whole thing is useless.