  We consider a broad class of systems of nonlinear integro-differential
equations posed on the real line that arise as Euler-Lagrange equations to
energies involving nonlinear nonlocal interactions. Although these equations
are not readily cast as dynamical systems, we develop a calculus that yields a
natural Hamiltonian formalism. In particular, we formulate Noether's theorem in
this context, identify a degenerate symplectic structure, and derive
Hamiltonian differential equations on finite-dimensional center manifolds when
those exist. Our formalism yields new natural conserved quantities. For
Euler-Lagrange equations arising as traveling-wave equations in gradient flows,
we identify Lyapunov functions. We provide several applications to
pattern-forming systems including neural field and phase separation problems.
