  We report on transcritical bifurcations of periodic orbits in non-integrable
two-dimensional Hamiltonian systems. We discuss their existence criteria and
some of their properties using a recent mathematical description of
transcritical bifurcations in families of symplectic maps. We then present
numerical examples of transcritical bifurcations in a class of generalized
H\'enon-Heiles Hamiltonians and illustrate their stabilities and unfoldings
under various perturbations of the Hamiltonians. We demonstrate that for
Hamiltonians containing straight-line librating orbits, the transcritical
bifurcation of these orbits is the typical case which occurs also in the
absence of any discrete symmetries, while their isochronous pitchfork
bifurcation is an exception. We determine the normal forms of both types of
bifurcations and derive the uniform approximation required to include
transcritically bifurcating orbits in the semiclassical trace formula for the
density of states of the quantum Hamiltonian. We compute the coarse-grained
density of states in a specific example both semiclassically and quantum
mechanically and find excellent agreement of the results.
