How to do the 'curvy L' for Lagrangian or Laplace Transforms?

You have been told how to get a curved L. But here's some more general advice, which also applies in this situation: In cases such as this, always create your own shortcut macro, say

newcommand{Lagr}{mathcal{L}}

This way, if you ever decide that that curly L is not quite the one you like, it is easy to switch. Also, even before you knew the answer to this question, you could have written

newcommand{Lagr}{L}

in your preamble. Then you could start/continue writing the document, use ordinary capital Ls where you want, and Lagr wherever you actually mean a Lagrangian, and then later changing the definition of Lagr to something appropriate. This way, you wouldn't have to manually search for each occurence of a capital L and decide if it's a Lagrangian or not. Clearly Lagr (or whatever you want to call this macro) is also easier to type than mathcal{L}, and it makes the source code much more readable.

Another advantage, which is probably more subtle, since you're new to LaTeX, is that we can make the curvy L exactly the type of math we want. TeX distinguishes between ordinary letters, binary operators, binary relations, opening and closing parenthesis and so on; the spacing between two ordinary letters is not the same as the spacing between the a, +, and b in $a+b$. So since the Lagrangian is a kind of operator, we could say

newcommand{Lagr}{mathop{mathcal{L}}}

But in the case of operators, the package amsmath (which you are most likely using; if not, you should) provides a somewhat better mechanism:

DeclareMathOperator{Lagr}{mathcal{L}}

Added: Another (related) tip: Even if you are using the same notation for two different things, it is best to make a separate macro for each. In this case you might have

DeclareMathOperator{Lagr}{mathcal{L}}
DeclareMathOperator{Lapl}{mathcal{L}}

The reason is the same as in the L vs. Lagr case above: If you at some point decide that using mathcal{L} for both is a bad idea, you would have to find each occurence of Lagr and figure out if it is really a Laplacian. Using macro names carrying semantic meaning is one of the great powers of TeX.