- #1
Spinny
- 20
- 0
Here's the problem. For a neutral vector field V_{\mu} we have the Lagrangian density
\mathcal{L} = -\frac{1}{2}(\partial_{\mu}V_{\nu})(\partial^{\mu}V^{\nu})+\frac{1}{2}(\partial_{\mu}V^{\mu})(\partial_{\nu}V^{\nu})+\frac{1}{2}m^2V_{\mu}V^{\mu}
We are then going to use the Euler-Lagrange equations to show that (for m\neq 0)
\partial_{\mu}\partial^{\mu}V^{\nu}+m^2V^{\nu} = 0 \quad;\quad \partial^{\mu}V_{\mu} = 0
Now, the Euler-Lagrange equation (as I found in the textbook) is
\frac{\partial \mathcal{L}}{\partial \varphi}-\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\varphi)} = 0
My problem here, with what at first glance would appear to be a rather simple problem, is that I'm confused by all the indices! I haven't a lot of experience working like this, in fact this is all new to me, so I don't quite know how or where to start.
I can, however, give a specific example of what I don't understand, just to get things started.
The first part seemed at first easy enough, as the Lagrangian only contains one part with V_{\mu} which is
\frac{1}{2}m^2V_{\mu}V^{\mu}
and from what I've understood V_{mu}V^{\mu} is just the square of each of the components of the vector, so that when you derivate it with respect to V_{\mu}, I thought you'd get something like
m^2V^{\mu}
but the text says m^2 V^{\nu} which brings up two question, first of all, why is it \nu and not \mu, and why is it an upper index, rather than a lower one?
Furthermore I was wondering if anyone could recommend a good book introductory book about tensor algebra and such, preferably one intended for physicists rather than mathematicians. (It doesn't have to be a book on just about tensors, as long as it contains a good introduction to tensors.)
\mathcal{L} = -\frac{1}{2}(\partial_{\mu}V_{\nu})(\partial^{\mu}V^{\nu})+\frac{1}{2}(\partial_{\mu}V^{\mu})(\partial_{\nu}V^{\nu})+\frac{1}{2}m^2V_{\mu}V^{\mu}
We are then going to use the Euler-Lagrange equations to show that (for m\neq 0)
\partial_{\mu}\partial^{\mu}V^{\nu}+m^2V^{\nu} = 0 \quad;\quad \partial^{\mu}V_{\mu} = 0
Now, the Euler-Lagrange equation (as I found in the textbook) is
\frac{\partial \mathcal{L}}{\partial \varphi}-\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\varphi)} = 0
My problem here, with what at first glance would appear to be a rather simple problem, is that I'm confused by all the indices! I haven't a lot of experience working like this, in fact this is all new to me, so I don't quite know how or where to start.
I can, however, give a specific example of what I don't understand, just to get things started.
The first part seemed at first easy enough, as the Lagrangian only contains one part with V_{\mu} which is
\frac{1}{2}m^2V_{\mu}V^{\mu}
and from what I've understood V_{mu}V^{\mu} is just the square of each of the components of the vector, so that when you derivate it with respect to V_{\mu}, I thought you'd get something like
m^2V^{\mu}
but the text says m^2 V^{\nu} which brings up two question, first of all, why is it \nu and not \mu, and why is it an upper index, rather than a lower one?
Furthermore I was wondering if anyone could recommend a good book introductory book about tensor algebra and such, preferably one intended for physicists rather than mathematicians. (It doesn't have to be a book on just about tensors, as long as it contains a good introduction to tensors.)