Deriving Commutation of Variation & Derivative Operators in EL Equation

• hideelo
In summary: This means that the variations of q(t) and \delta q(t) can be treated as independent variables in the derivation of the Euler-Lagrange equations.In summary, the principle of least action involves the equality of variations in functions q(t) and q'(t), which is justified by treating q(t) and \delta q(t) as independent variables and not varying time or the endpoints of the action integral.
hideelo
I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality

δ(dq/dt) = d(δq)/dt

Where q is some coordinate, and δf is the first variation in f. In general, this can be seen more broadly, given a scalar field ψ

δ(∂ψ/∂x) = ∂(δψ)/∂x

Where x is any independant variable (i.e. x,y,z,t or any other coordinate system)

How are these equalities justified?

hideelo said:
I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality

δ(dq/dt) = d(δq)/dt

Where q is some coordinate, and δf is the first variation in f. In general, this can be seen more broadly, given a scalar field ψ

δ(∂ψ/∂x) = ∂(δψ)/∂x

Where x is any independant variable (i.e. x,y,z,t or any other coordinate system)

How are these equalities justified?

Well, if you have two functions $q(t)$ and $q'(t)$ that are related by:

$q'(t) = q(t) + \delta q(t)$

Then clearly:

$\dfrac{d q'}{dt} = \dfrac{dq}{dt} + \dfrac{d \delta q}{dt}$

So if $\delta \dfrac{dq}{dt}$ is defined to be $\dfrac{dq'}{dt} - \dfrac{dq}{dt}$, then the conclusion follows.

I suppose there are different ways to think about it, but I don't think of $\delta$ as being an operator. $\delta q$ is just a function of $t$, that's assumed to be small.

vanhees71 and hideelo
Another important point is that time (and also the endpoints of the action integral) are not varied in Hamilton's principle.

1. What is the purpose of deriving commutation of variation and derivative operators in EL equation?

The purpose of deriving commutation of variation and derivative operators in EL equation is to understand the relationship between variations in a functional and the corresponding variations in its derivative. This is important in optimization and variational calculus, where these operations play a crucial role in finding the extrema of a functional.

2. How do you derive the commutation of variation and derivative operators?

The commutation of variation and derivative operators can be derived by using the chain rule and the fundamental theorem of calculus. This involves taking the derivative of the functional with respect to the original variable and then swapping the order of differentiation and integration.

3. Can you explain the concept of variation in EL equation?

Variation in EL equation refers to the change in a functional due to small changes in its input variables. This is represented by the Greek letter delta (∆) and is used to find the extrema of a functional by setting the variation to zero.

4. What is the importance of commutation of variation and derivative operators in physics?

In physics, commutation of variation and derivative operators is important for solving problems in classical mechanics, quantum mechanics, and field theory. These operations are used to derive the equations of motion and determine the stationary points of a system, which are crucial in understanding the behavior of physical systems.

5. Are there any real-world applications of commutation of variation and derivative operators?

Yes, there are many real-world applications of commutation of variation and derivative operators. Some examples include determining the optimal shape of a bridge to minimize stress, finding the path of least resistance for a river, and minimizing energy consumption in electronic circuits. These operations are also used in economics, biology, and engineering for optimization problems.

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