Variational derivative and Euler-Poincare equations

In summary, Darryl Holm uses a variational derivative in his book "Geometric mechanics and symmetry". The Euler-Poincare reduction theorem states that the variational derivative reduces to partial derivatives of ##l##.
  • #1
eipiplusone
10
0
Hi,

I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", where he suddenly uses what seems to be a variational derivative, which I'm having a hard time understanding. The Euler-Poincare reduction theorem (and equation) goes as follows:
upload_2019-2-19_21-5-6.png


It is the ##\frac{\delta l}{\delta \xi}## - derivative which I'm guessing is a variational derivative. It must be a covector, and in computations it seems to reduce to partial derivatives of ##l## wrt. ##\xi## (in suitable coordinates) - I have also seen other sources where the EP-equation is stated in terms of ##\frac{\partial l}{\partial \xi}## instead of the abovementioned.

He defines it in two different ways. The first definition is found in one of the later chapters of the book (and it seems to be in a more general setting):

upload_2019-2-19_21-10-26.png

And in another book ("Geometric mechanics - part 2") he defines it as:

upload_2019-2-19_20-53-28.png


My questions are:

- which definition should I focus on?
- Is the pairing in the latter definition the usual covector-vector pairing?

Any explanations, hints or references would be greatly appreciated! Thanks.

<mentor: fix latex>
 

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  • #2
eipiplusone said:
My questions are:

- which definition should I focus on?
##11.13## The other definition ##(2.1.1)## is basically just the definition of a derivative. The crucial point is what is variated, namely the direction ##v:=\delta q##, resp. ##v:=\delta u##.
You could also combine them and get both in one:
$$
\delta l[ u ]= \lim_{t \to 0}\dfrac{l[ u+t\delta u ] - l[ u ]}{t}=\left. \dfrac{d}{dt}\right|_{t=0}l[ u+t\delta u ]=\left\langle \dfrac{\delta l}{\delta u},\delta u \right\rangle = \int \dfrac{\delta l}{\delta u} \cdot \delta u \,dV
$$
- Is the pairing in the latter definition the usual covector-vector pairing?
Yes. It's the view of the derivative as a linear function, similar to the gradient in ##\mathbb{R}^n##: ##d l(u)(v)=\langle \nabla l(u) , v \rangle##, except that we do not take the total differential here but only the partial along ##\delta u##.
 
  • #3
Thanks for your answer. I am still confused though :/
  • I don't understand the last equality; ##\langle \frac{\delta l}{\delta u} , \delta u \rangle = \int \frac{\delta l}{\delta u} \cdot \delta u \hspace{1mm} d V ##. I would think that ## \frac{\delta l}{\delta u} \cdot \delta u ## is the euclidean inner product representation of the evaluation ## \frac{\delta l}{\delta u}[\delta u] \in \mathbb{R}##, where ##\frac{\delta l}{\delta u}, \delta u## are euclidean vectors. But in that case the integral doesn't make sense, since ##\langle \frac{\delta l}{\delta u} , \delta u \rangle = \frac{\delta l}{\delta u} \cdot \delta u ## . So... why the need for the integral? I'm probably missing something fundamental.
  • Is it correct to say that if ##l## is an ordinary function of a vector ##u##, then ##\frac{\delta l}{\delta u} = \frac{\partial l}{\partial u} ##?
 
  • #4
eipiplusone said:
Thanks for your answer. I am still confused though :/
  • I don't understand the last equality; ##\langle \frac{\delta l}{\delta u} , \delta u \rangle = \int \frac{\delta l}{\delta u} \cdot \delta u \hspace{1mm} d V ##. I would think that ## \frac{\delta l}{\delta u} \cdot \delta u ## is the euclidean inner product representation of the evaluation ## \frac{\delta l}{\delta u}[\delta u] \in \mathbb{R}##, where ##\frac{\delta l}{\delta u}, \delta u## are euclidean vectors. But in that case the integral doesn't make sense, since ##\langle \frac{\delta l}{\delta u} , \delta u \rangle = \frac{\delta l}{\delta u} \cdot \delta u ## . So... why the need for the integral? I'm probably missing something fundamental.
The integral comes from the definition of the inner product in ##L^2(I)\, : \,\langle f,g \rangle = \int_I f(x)g^*(x)dx## only that ##f,g## in our case aren't functions but differential forms, and ##I## not an interval but a vector field ##V##.
  • Is it correct to say that if ##l## is an ordinary function of a vector ##u##, then ##\frac{\delta l}{\delta u} = \frac{\partial l}{\partial u} ##?
Yes. However, partials are usually meant as coordinate directions, and ##u## doesn't have to be one.

The main difficulty, at least mine, is to distinguish the roles a derivative is playing in a certain context. I've made the fun and listed a couple (10) of them here: https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ but this has meant to be more as a hint on how easy it is to get confused rather than an explanation. Fun fact: slope wasn't even on the list.

My attempt to sort things out was this one: https://www.physicsforums.com/insights/pantheon-derivatives-part-ii/#toggle-id-0
If you look at the pdf, then it's probably better to read, resp. easier to search for keywords, e.g. Gâteaux or Noether as in our case here.
 
  • #5
Thank you. The documents you link to looks very nice, I will see if they can make things clear to me.
 

1. What is a variational derivative?

A variational derivative is a mathematical tool used in the calculus of variations to find the functional derivative of a functional with respect to its independent variables. It is similar to a partial derivative, but instead of taking the derivative with respect to a single variable, it takes the derivative with respect to a function.

2. What are Euler-Poincare equations?

The Euler-Poincare equations are a set of equations that describe the dynamics of a system in terms of its variational derivatives. They are derived from the principle of least action, which states that the path a system takes between two points in time is the one that minimizes the action functional.

3. How are variational derivatives and Euler-Poincare equations related?

Variational derivatives play a crucial role in the derivation of the Euler-Poincare equations. The equations are derived by taking the variational derivative of the Lagrangian of a system with respect to its independent variables. This results in a set of differential equations that describe the dynamics of the system.

4. What are some applications of variational derivative and Euler-Poincare equations?

Variational derivative and Euler-Poincare equations have various applications in physics, engineering, and mathematics. They are used to solve problems in mechanics, fluid dynamics, and control theory. They are also used in the study of geodesics, optimal control, and shape optimization.

5. Are there any limitations to using variational derivative and Euler-Poincare equations?

While variational derivative and Euler-Poincare equations have a wide range of applications, they are not suitable for all types of systems. They are most commonly used for systems that can be described by a Lagrangian, and may not be applicable to systems with complex or nonlinear dynamics. Additionally, the derivation of these equations can be mathematically challenging and may require specialized knowledge and techniques.

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