Recently I installed Ubuntu 14.10 on a 2TB drive, because it was handy. I selected the LVM filesystem because I thought it might be good to learn more about it. I know virtually nothing about it now, though.

I discovered that I could not back up this installation with the software I had been using. I made an identical installation on a 500GB drive, and found that Clonezilla would back it up. Clonezilla will not copy an image made from the 2TB drive to the smaller one, but I suppose it would copy it to another 2TB drive if necessary, so the backup problem is solved.

At the same time, I'd like to free up the 2TB drive, since I don't need that much space for the Ubuntu installation. The partition structure on the 500GB is probably the same (and, perhaps unfortunately, the volume names are also the same). So it occurred to me that an application such as rsync might be appropriate to copy the files recursively from the large drive to the small one. Could anyone give me advice about how to do this (preferably avoiding the use of the term "simply...")?

There are a lot of "non simple" strategies - most people try to avoid them and go for the "simply <blah>" ones ...

Some more detail - if you have two identical installs why do you need to copy anything ?. More pertinently, what do you really want/need to copy ?. User data, system (why), both ?.
If as you suspect both installs have the same vg and/or lv name(s) you can''t mount both concurrently - so any "simple" copy is out of he question unless one is renamed. Let's see the following to start with (both disks) - and use [code] tags please

Both installs of Ubuntu 14.10 (x64) were made using all default options, but one was on a 2TB disk and the other on a 500GB disk. However, many modifications were made to the 2TB disk before its replacement by the smaller disk was contemplated. They both have the same partition structure, though the amount of free space is vastly different, leading me to believe that copying the files with a utility like rsync would be appropriate. Elsewhere someone recommended Dervish, which is based on rsync, and perhaps this is what I should use.

"Recently I installed Ubuntu 14.10 on a 2TB drive, "

I'd just do a clean new install. Can't have gotten too used to it.