5 ms·

But are the statements "reality minimizes action" and "reality solves differential equations" actually, literally equivalent to each other? It strikes me that t

But are the statements "reality minimizes action" and "reality solves differential equations" actually, literally equivalent to each other? It strikes me that this is not in fact the case. There are solutions to systems describing skyscraper sway that have the building stand at a terrifying lean angle; what is supposed to be selecting the particular solutions that we observe? A purely random walk through the solution space seems unsatisfying as an answer.

whatshisface 7y ago
They are actually, literally equivalent to each other. Differential equations aren't a "purely random walk," they are a statement of what is about to happen based on what is happening. You have three formalisms: Newtonian (differential equations), Lagrangian (action minimization), and Hamiltonian (differential equations, but derived from a scalar field kind of like in Lagrangian mechanics). They are all different ways of writing down the same thing, and their advantages and disadvantages are related to situational convenience.
- Brain_Thief 7y ago
  Thank you for the response, I find this very interesting. So is it true to say that the production of a particular solution to a differential equation is a statement about how a system will behave based on its initial conditions, and that the statement captures within it the principle of action minimization by virtue of the fact that it is a derivation of information from natural laws?
  - whatshisface 7y ago
    "Action is minimized" and "F = ma" are the same in the sense that x+5=0 and x+6=1 are the same. In both cases you can go from one to the other with a sequence of deductions. There's no way to tell which one is the "natural law" because you can take either one as a fact and derive the other as a consequence.
    - Brain_Thief 7y ago
      Interesting - your response makes sense. Thanks again for the information.

whatshisface 7y ago

They are actually, literally equivalent to each other. Differential equations aren't a "purely random walk," they are a statement of what is about to happen based on what is happening. You have three formalisms: Newtonian (differential equations), Lagrangian (action minimization), and Hamiltonian (differential equations, but derived from a scalar field kind of like in Lagrangian mechanics). They are all different ways of writing down the same thing, and their advantages and disadvantages are related to situational convenience.

Brain_Thief 7y ago
Thank you for the response, I find this very interesting. So is it true to say that the production of a particular solution to a differential equation is a statement about how a system will behave based on its initial conditions, and that the statement captures within it the principle of action minimization by virtue of the fact that it is a derivation of information from natural laws?
- whatshisface 7y ago
  "Action is minimized" and "F = ma" are the same in the sense that x+5=0 and x+6=1 are the same. In both cases you can go from one to the other with a sequence of deductions. There's no way to tell which one is the "natural law" because you can take either one as a fact and derive the other as a consequence.
  - Brain_Thief 7y ago
    Interesting - your response makes sense. Thanks again for the information.