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.

$\mathcal{L}$

Have a look at “How to look up a math symbol?” for ideas how you can easily find a particular symbol.

You can also use the mathrsfs package (put \usepackage{mathrsfs} in your preamble), and its command \mathscr (e.g. $\mathscr{L}$ should give you what you want) to get more script-like, curvy letters.