Coupled ODEs from Euler Lagrange eq

In summary, the conversation discusses a differential equation given by $F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}$ and the corresponding Euler-Lagrange equations. It is noted that the equation can be simplified by defining $v_1 = u_1'$ and $v_2 = u_2'$, which transforms it into an algebraic equation. It is also mentioned that the equation is relatively easy to solve.
  • #1
Dustinsfl
2,281
5
Given \(F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}\).
\[
\frac{\partial F}{\partial u_i} - \frac{d}{dx}\left[\frac{\partial F}{\partial u_i'}\right] = 0
\]
From the E-L equations, I found
\begin{align*}
\frac{d}{dx}\left[2Au_1' + Bu_2'\right] &= 0\\
\frac{d}{dx}\left[2Cu_2' + Bu_1'\right] &= 0\\
2Au_1' + Bu_2' &= D\\
2Cu_2' + Bu_1' &= E
\end{align*}

Just a general question. Is this DE going to be extremely difficult to solve or is it relativily trivial?
 
Physics news on Phys.org
  • #2
It's going to be relatively trivial. Let $v_{1}=u_{1}'$ and $v_{2}=u_{2}'$. Then it's not actually a DE in the $v_{i}$'s, but just an algebraic equation. This happens because there are no $u_{i}$ terms, but only their derivatives. So you can solve for the $v_{i}$'s using your favorite method, and then at least symbolically integrate the result.
 

FAQ: Coupled ODEs from Euler Lagrange eq

What are coupled ODEs in the context of Euler-Lagrange equations?

Coupled ODEs, or coupled ordinary differential equations, refer to a system of differential equations that are interdependent. In the context of Euler-Lagrange equations, coupled ODEs arise when there are multiple variables in the system that are related through the Lagrangian function. This means that the equations of motion for each variable are connected and must be solved simultaneously.

How are coupled ODEs derived from Euler-Lagrange equations?

Coupled ODEs can be derived from the Euler-Lagrange equations by applying the principle of least action. This principle states that the true path of a system between two points is the one for which the action (defined by the integral of the Lagrangian over time) is stationary. By varying the path and setting the resulting variations to zero, the Euler-Lagrange equations can be obtained, which are a set of coupled ODEs.

What are the applications of coupled ODEs from Euler-Lagrange equations?

Coupled ODEs from Euler-Lagrange equations have many applications in physics and engineering. They are commonly used in classical mechanics to model the motion of systems with multiple degrees of freedom, such as a double pendulum. They are also used in fields such as fluid dynamics, quantum mechanics, and control systems.

How are coupled ODEs solved?

There are various numerical methods that can be used to solve coupled ODEs, including the Euler method, Runge-Kutta methods, and the shooting method. These methods involve breaking the system of equations into smaller, simpler equations that can be solved iteratively. There are also some analytical techniques, such as separation of variables and Laplace transforms, that can be used for certain types of coupled ODEs.

What are some challenges associated with solving coupled ODEs from Euler-Lagrange equations?

Solving coupled ODEs from Euler-Lagrange equations can be challenging due to the interdependence of the equations and the complexity of the system being modeled. The system may also have multiple solutions or exhibit chaotic behavior, making it difficult to find an exact solution. Additionally, numerical methods may introduce errors and require careful selection of step size and other parameters for accurate results.

Similar threads

I The Euler-Lagrange equation and the Beltrami identity
Replies
1
Views
2K
I Second Order ODE with Exponential Coefficients
Replies
1
Views
2K
How to derive the associated Euler-Lagrange equation?
Replies
6
Views
1K
Modifying Euler-Lagrange equation to multivariable function
Replies
1
Views
2K
Deriving ODEs for straight lines in polar coordinates for a given Lagrangian
Replies
8
Views
919
MHB Solving Rational ODE's of the form (ax+by+c) dx+(ex+fy+g) dy=0.
Replies
1
Views
5K
MHB How do I solve the Euler-Lagrange equation for this functional?
Replies
1
Views
1K
I Numerically solving a non-local PDE
Replies
3
Views
2K
I Nonlinear Second Order ODE: Can We Find an Analytical Solution?
Replies
2
Views
3K
I Solving an ODE with Legendre Polynomials
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top