DOTDO
- 8
- 2
Hi.
I have thought that the variables q and q' in L = L(q, q') are independent. (q' = dq/dt)
Of course q and q' are functions of time t , but they are only dependent in terms of t .
However, in the sight of general(or abstract? I mean, not specific) functional L(q, q'),
q and q' are just independent variables of L .
Only after L is determined by using calculus of variations, they are dependent.
Here is more detailed logic of mine.
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1a) Think about the coordinates (x, y, z). All x, y and z are independent thus they can make the coordinates. And there are infinite possible functions z = f(x,y).
2a) Let's choose a specific function among those infinite possible functions, say, a line (x, y, z) = (s, s, s)
where s is a parameter. Then, by considering this function, x and y are no longer independent because
x=y on this line.
1b) There is a coordinates (q, q', L). There are infinite possible functions L = L(q, q').
And We don't know what path the particle will follow. When q=1, q' can be 1 or 100. When q=2, q' can be 1.1 or 500. This means that there are infinite possible paths, or physical laws.
On each path q and q' are dependent. But considering all the paths, eventually it means q and q' are independent variables.
2b) In 2a, we chose a specific function. Here, in Lagrangian, it is same to adding a condition, the principle of least action, which yields the Euler-Lagrange equation.
By this condition, a specific Lagrangian L is chosen, which makes q and q' dependent.
And this also means that among those infinite possible paths, we chose a specific path, which can be said that we chose a specific law, Newton's 2nd law, among those infinite possible physical laws.
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Is this logic wrong or not?
Please point out where I am wrong. Thank you.
I have thought that the variables q and q' in L = L(q, q') are independent. (q' = dq/dt)
Of course q and q' are functions of time t , but they are only dependent in terms of t .
However, in the sight of general(or abstract? I mean, not specific) functional L(q, q'),
q and q' are just independent variables of L .
Only after L is determined by using calculus of variations, they are dependent.
Here is more detailed logic of mine.
-------------------------------------------------------------------------------
1a) Think about the coordinates (x, y, z). All x, y and z are independent thus they can make the coordinates. And there are infinite possible functions z = f(x,y).
2a) Let's choose a specific function among those infinite possible functions, say, a line (x, y, z) = (s, s, s)
where s is a parameter. Then, by considering this function, x and y are no longer independent because
x=y on this line.
1b) There is a coordinates (q, q', L). There are infinite possible functions L = L(q, q').
And We don't know what path the particle will follow. When q=1, q' can be 1 or 100. When q=2, q' can be 1.1 or 500. This means that there are infinite possible paths, or physical laws.
On each path q and q' are dependent. But considering all the paths, eventually it means q and q' are independent variables.
2b) In 2a, we chose a specific function. Here, in Lagrangian, it is same to adding a condition, the principle of least action, which yields the Euler-Lagrange equation.
By this condition, a specific Lagrangian L is chosen, which makes q and q' dependent.
And this also means that among those infinite possible paths, we chose a specific path, which can be said that we chose a specific law, Newton's 2nd law, among those infinite possible physical laws.
-------------------------------------------------------------------------------
Is this logic wrong or not?
Please point out where I am wrong. Thank you.