  Classical Hamiltonian systems with conserved charges and those with
constraints often describe dynamics on a pre-symplectic manifold. Here we show
that a pre-symplectic manifold is also the proper stage to describe autonomous
energy conserving Hamiltonian time crystals. We explain how the occurrence of a
time crystal relates to the wider concept of spontaneously broken symmetries;
in the case of a time crystal, the symmetry breaking takes place in a dynamical
context. We then analyze in detail two examples of time crystalline Hamiltonian
dynamics. The first example is a piecewise linear closed string, with dynamics
determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane
stability. We explain how the Lie-Poisson brackets descents to a time
crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics
supports two phases; in one phase we have a time crystal and in the other phase
time crystals are absent. The second example is a discrete Hamiltonian variant
of the Q-ball Lagrangian of time dependent non-topological solitons. We explain
how a Q-ball becomes a time crystal, and we construct examples of time
crystalline Q-balls.
