How to Prove Lagrange's Equations for Shallow Water using Chain Rule?

[bigevil]

Homework Statement

In a shallow layer of water, the velocity of water in the z direction may be ignored and is therefore $$(\dot{x},\dot{y})$$. We can define the Lagrangian coordinates such that the depth of water h is satisfied by the relations

Given that $$h = \frac{1}{\alpha}$$ and $$\alpha = \frac{\partial(x,y)}{\partial(a,b)}$$

and the Lagrangian density is given as

$$L = \frac{1}{2}\dot{x}^2 + \frac{1}{2}\dot{y}^2 - \frac{1}{2}gh(x_a,x_b,y_a,y_b)$$

where $$p_q = \frac{\partial p}{\partial q}$$.

Given further that Lagrange's equations for a 2D continuous system are known to be

$$\frac{D}{Dt}\left( \frac{\partial L}{\partial\dot{x}} \right) + \frac{\partial}{\partial a} \left( \frac{\partial L}{\partial x_a} \right) + \frac{\partial}{\partial b} \left( \frac{\partial L}{\partial x_b} \right) - \frac{\partial L}{\partial x} = 0$$

with a similar equation for the y variable, prove that

$$\frac{D\dot{x}}{Dt} + g \frac{\partial h}{\partial x} = 0$$

Homework Equations

I know the general approach of this problem, but my main problem comes in substituting

$$\frac{\partial}{\partial a} \frac{\partial L}{\partial x_a}$$.

If I apply chain rule on the Lagrangian here,
$$\frac{\partial}{\partial a} \frac{\partial L}{\partial h} \frac{\partial h}{\partial x_a} = -\frac{g}{2} \frac{\partial}{\partial a} \frac{\partial h}{\partial x_a}$$
How do I proceed after this point?

 

[mark726]

Thank you for your post. It seems like you are on the right track with your approach to this problem. In order to continue, you will need to use the chain rule again to expand the term \frac{\partial}{\partial a} \frac{\partial L}{\partial x_a} , taking into account that h = \frac{1}{\alpha} and \alpha = \frac{\partial(x,y)}{\partial(a,b)}. This will allow you to simplify the expression and eventually arrive at the desired result.

Another helpful tip is to keep in mind that the Lagrangian density L is a function of x, y, \dot{x}, and \dot{y}. This will help you determine which variables to take derivatives with respect to and how to apply the chain rule.

I hope this helps in solving the problem. Good luck!