Are q and q' dependent variables in Lagrangian or not?

In summary: You have to find the relationship between those two to get the equation of motion.In summary, the conversation discusses the independence of the variables q and q' in the Lagrangian L = L(q, q') and how they become dependent when considering specific functions or paths. The concept of Lagrangian and Euler-Lagrange equations is also mentioned, and it is noted that the independent variables are q and q'. The logic presented is deemed correct, and the example of a mass on a spring is used to further illustrate the concept of independent variables.
  • #1
DOTDO
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0
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.
-------------------------------------------------------------------------------

Is this logic wrong or not?

Please point out where I am wrong. Thank you.
 
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  • #2
Logic is just fine. Lagrangian has independent variables ##q## and ##\dot q## (and possibly others). Euler-Lagrange equations establish a relationship and you get the equations of motion.

Compare the mechanical energy of a mass on a spring: E is a sum of spring energy, gravitational potential energy and kinetic energy. So the independent variables are ##y## and ##\dot y##.
 
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1. Are q and q' dependent variables in Lagrangian or not?

Yes, q and q' are dependent variables in Lagrangian. This means that the value of one variable is affected by the value of the other variable.

2. What is the significance of q and q' being dependent variables in Lagrangian?

The dependence of q and q' in Lagrangian equations allows for the prediction and understanding of the dynamics of a system. By knowing the values of q and q', the behavior and motion of a system can be determined.

3. How are q and q' related in Lagrangian equations?

Q and q' are related through the Lagrangian function, which represents the total energy of a system. The derivative of the Lagrangian with respect to q' is equal to the momentum, and the derivative of the Lagrangian with respect to q is equal to the force acting on the system.

4. Can q and q' be considered independent variables in Lagrangian?

No, q and q' cannot be considered independent variables in Lagrangian. In order for Lagrangian equations to accurately describe a physical system, q and q' must be dependent on each other.

5. Are q and q' the only dependent variables in Lagrangian equations?

No, in addition to q and q', there may be other dependent variables in Lagrangian equations that represent different physical quantities, such as potential energy or angular momentum.

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