Euler Lagrange Equation



Euler-Lagrange Equation for $\int F(x, y, y’) dx$:

$$\frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = \frac{\partial F}{\partial y}$$

– Dependent variable missing, E-L equation becomes:

$$\frac{\partial F}{\partial y’} = constant$$

– Independent variable missing, E-L equation becomes:

$$F – y’\frac{\partial F}{\partial y’} = constant$$

_____________

Several dependent variables:

E.g. $F = F (x, y_1 , y_2 , … , y_n , y’_1 , … , y’_n )$

$$\frac{\partial F}{\partial y_i} = \frac{d}{dx}\left( \frac{\partial F}{\partial y’_i} \right)$$

Several independent variables:

$$\frac{\partial F}{\partial y} = \sum_{i=1}^{n} \frac{\partial}{\partial x_i} \left( \frac{\partial F}{\partial y_{x_{i}}}\right), y_{x_{i}} = \frac{\partial y}{\partial x_i}$$

____________

Variable end points:

$$I = \int^b _a F(x,y,y’) dx$$

If lower end point a is fixed, we require:

$$\left. \frac{\partial F}{\partial y’} \right|_{x = b} = 0$$

If both ends can vary, we require:

$$\left. \frac{\partial F}{\partial y’} \right|_{x = a} = 0,$$

$$\left. \frac{\partial F}{\partial y’} \right|_{x = b} = 0$$

______________

Constraint

$$I = \int^b _a F(x,y,y’) dx$$

Constraint: $J = \int^b _a G(x,y,y’) dx$

\begin{array} {lcl} K & = & I + \lambda J \\ & = & \int^b _a F + \lambda G dx \end{array}

 

Back To Mathematics For An Undergraduate Physics Course



Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.