# Euler Lagrange equations in continuum

• JD_PM
In summary,The author is discussing a problem and feels lost. They ask for help from a colleague, and eventually realize that they need to discretize the terms in equation 5. They are still not sure why some terms should vanish when taking the limit.
JD_PM
Homework Statement
Given

$$\frac{d}{dt} \frac{\partial L}{\partial \dot \phi_a^{(i j k)}} - \frac{\partial L}{\partial \phi_a^{(i j k)}} = 0 \tag{1}$$

Show that, in the limit ##l \rightarrow 0##, we obtain

$$\partial_{\mu} \frac{\partial \mathscr{L}}{\partial(\partial_{\mu} \phi_a)} - \frac{\partial \mathscr{L}}{\partial \phi_a} = 0 \tag{2}$$
Relevant Equations
OK I've been stuck for a while in how to derive ##(1)##, so I better solve a simplified problem first:

We work with

Where

$$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$

And ##(3)## implies that ##\mathscr{L}(\vec x, t)##

We know that

$$L=l^3\sum_{(i j k)} \mathscr{L}^{(i j k)}(t) \tag{4}$$

Where

$$\lim_{l \rightarrow 0} L = \int d^3 \vec x \mathscr{L} \tag{5}$$

So, in analogy with ##(3), \mathscr{L}^{(i j k)}## depends on the fields ##\phi_a^{(i j k)} (t)##, on the time derivative of the fields ##\dot \phi_a^{(i j k)} (t)## and on the partial derivative of the fields with respect to ##x, y## and ##z## i.e;

$$\frac{\phi^{(i+1, j, k)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.1}$$

$$\frac{\phi^{(i, j+1, k)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.2}$$

$$\frac{\phi^{(i, j, k+1)}(t)-\phi^{(i, j, k)}(t)}{l} \tag{6.3}$$Let's tackle the problem.

I would naively plug ##(4)## into ##(1)## and evaluate the terms to get

$$\frac{\partial}{\partial \dot \phi_a^{(i j k)}} \sum_{(i'j'k')} \Big[ l^3 \mathscr{L}^{(i' j' k')}(t) \Big] = l^3 \frac{\partial \mathscr{L}^{(i j k)}}{\partial \dot \phi_a^{(i j k)}} \tag{7.1}$$

$$\frac{\partial}{\partial \phi_a^{(i j k)}} \sum_{(i'j'k')} \Big[ l^3 \mathscr{L}^{(i' j' k')}(t) \Big] = l^3 \frac{\partial \mathscr{L}^{(i j k)}}{\partial \phi_a^{(i j k)}} \tag{7.2}$$

Where I've used the Kronecker delta.

Next I'd take the limit ##l \rightarrow 0##\$ of ##\frac{\partial \mathscr{L}^{(i j k)}}{\partial \dot \phi_a^{(i j k)}}## and ##\frac{\partial \mathscr{L}^{(i j k)}}{\partial \phi_a^{(i j k)}}## to get ##\frac{\partial \mathscr{L}}{\partial \dot \phi_a}## and ##\frac{\partial \mathscr{L}}{\partial \phi_a}## (respectively)

So I get

$$l^3 \Big( \frac{d}{dt} \frac{\partial \mathscr{L}}{\partial \dot \phi_a} - \frac{\partial \mathscr{L}}{\partial \phi_a} \Big) = 0 \tag{8}$$

Which of course does not yield ##(2)## when taking the limit ##l \rightarrow 0##.

The issue is that I am missing the spatial components ##(6.1), (6.2), (6.3)##...

I've been discussing this problem and related but we did not manage to really understand it.

Any help is really appreciated.

Thank you

Delta2
Alright here's another attempt (after discussing with a colleague).

So it seems we are closer but we still do not see why certain terms in above's EQ. 5 should vanish when taking the limit ##l \rightarrow 0##.

The issue we really have is that we do not fully understand how to deal with discretized terms...

@samalkhaiat , would you have time for this?

etotheipi

## 1. What are Euler Lagrange equations in continuum?

Euler Lagrange equations in continuum are a set of mathematical equations that describe the behavior of continuous systems, such as fluids or deformable solids. They are derived from the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action, a measure of the system's energy.

## 2. How are Euler Lagrange equations used in physics?

Euler Lagrange equations are used to determine the equations of motion for a continuous system. They are commonly used in classical mechanics, fluid dynamics, and continuum mechanics to model and analyze the behavior of physical systems.

## 3. What is the difference between Euler Lagrange equations and Newton's laws of motion?

The main difference between Euler Lagrange equations and Newton's laws of motion is the approach to describing the behavior of a system. Newton's laws use the concept of forces to determine the motion of a system, while Euler Lagrange equations use the principle of least action. Additionally, Euler Lagrange equations are more general and can be applied to systems with complex geometries and constraints.

## 4. How are Euler Lagrange equations derived?

Euler Lagrange equations are derived by applying the calculus of variations to the action functional, which is a mathematical expression that represents the total energy of a system. The resulting equations are a set of partial differential equations that describe the dynamics of the system.

## 5. What are some applications of Euler Lagrange equations?

Euler Lagrange equations have a wide range of applications in physics and engineering. They are used to study the motion of fluids in pipes and channels, the deformation of structures under stress, and the behavior of electromagnetic fields. They also have applications in control theory, optimal control, and optimal design problems.

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