Blender was built and linked against the library named libavcodec-ffmpeg.so.56
which isn't there. The 56 at the end is the version which is one older than what you now have installed.
You can try creating a link with that name that points to the v57 version you have installed. ln -s /usr/local/lib/libavcodec-ffmpeg.so.57 /usr/local/lib/libavcodec-ffmpeg.so.56
. If you don't have the exact filename libavcodec-ffmpeg.so.57
try something similar, maybe libavcodec-extra.so.57
For small version changes that may work, larger variations are more likely to fail.
Find a third party repository that has blender built against libavcodec v57. Or one that has libavcodec v56.
If you can find a copy of libavcodec-ffmpeg.so.56
and don't want to replace over your existing libavcodec version, you should be able to create a lib dir next to the blender binary and place it in there, this will be easier if you have a self contained copy of blender rather than a system installed copy.
If you do have libavcodec-ffmpeg.so.56
installed then it may be a case of not being able to find it, setting LD_LIBRARY_PATH
in your environment can fix that.
Build blender yourself, which will always be linked with libraries you have installed.
libavcodec(-extra)57
: packages.ubuntu.com/zesty/blender , while xenial blender is linked to / deps onlibavcodec-ffmpeg(-extra)56
: packages.ubuntu.com/xenial/blender ; not sure why it is happening to you though