Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function $L$ is called the Lagrangian. In the case when $M$ is a Riemannian manifold and a particle in $M$ is moving under some conservative force field, taking the Lagrangian to be the kinetic minus the potential energy we recover Newton's second law.
I know that one of the main advantages to the Euler-Lagrange equations over Newtons is the way in which they simplify constrained systems. I know another is the coordinate independence of the equations. However, in all applications I've seen the manifold is always Riemannian and the Lagrangian is always $K-U$.
My questions are:
Why do we have this abstract definition of a Lagrangian system and of an abstract Lagrangian?
What are some of the cases in which $L$ is not $K-U$ that gives interesting results?
Or cases in which the manifold is not Riemannian?