The authors develop a general framework for the stability analysis of solitary waves in Hamiltonian partial differential equations posed on the real line. Main result is an instability criterion based on the sign of the derivative of the momentum of the solitary wave with respect to its speed of propagation. The stability problem is reduced to a family of nonautonomous, linear ordinary differential equations on the real line, depending on a spectral parameter \(\lambda\). Eigenvalues occur for values of \(\lambda\) where this family of differential equations possesses nontrivial, bounded solutions. J. W. Evans [Indiana Univ. Math. J. 24, 1169-1190 (1975; Zbl 0317.92006)] and J. Alexander, R. Gardner and C. Jones [J. Reine Angew. Math. 410, 167-212 (1990; Zbl 0705.35070)] constructed a complex analytic function \(E(\lambda)\) as a suitable determinant, whose zeroes coincide with eigenvalues. Here, this construction is clarified in the context of Hamiltonian partial differential equations, exhibiting the relation to temporal and spatial symplectic forms. Finally, the results are applied to solitary waves in a Boussinesq model, a nonlinear Schrödinger equation, and a complex nonlinear wave equation.