a Hamiltonian vector field
Take an energy function H on phase space — say the total energy of a pendulum as a function of its angle and momentum. How does the system actually move? In ordinary calculus the gradient of a function points 'uphill', and following it makes a function increase. On a symplectic manifold there is a twisted cousin of the gradient: instead of flowing uphill, it flows along the level sets, keeping H constant. That twisted gradient is the Hamiltonian vector field, and following it for time t is exactly the motion of the mechanical system with energy H.
Precisely, given a smooth function H on a symplectic manifold (M, omega) — the Hamiltonian — its Hamiltonian vector field X_H is the unique vector field defined by the equation iota_{X_H} omega = dH, that is omega(X_H, .) = dH(.). Nondegeneracy of omega is exactly what makes X_H exist and be unique (you can 'invert' omega to turn the 1-form dH into a vector field). In Darboux coordinates (q, p) with omega = sum dp_i ^ dq_i, this unpacks into Hamilton's equations: dq_i/dt = partial H / partial p_i and dp_i/dt = -partial H / partial q_i. The flow of X_H is the time evolution of the system, and it is automatically a family of symplectomorphisms.
Two facts make X_H special and worth keeping straight. First, energy is conserved along the motion: X_H(H) = dH(X_H) = omega(X_H, X_H) = 0 by antisymmetry, so H is constant on every trajectory — this is conservation of energy, falling straight out of the definition. Second, X_H is a stronger notion than a symplectic vector field. A vector field X is symplectic if its flow preserves omega, i.e. the Lie derivative L_X omega = 0, equivalently iota_X omega is closed; it is Hamiltonian if iota_X omega is exact (= dH for a globally defined H). On a simply connected manifold the two coincide, but in general the gap is measured by H^1(M; R).
The harmonic oscillator has H = (1/2)(q^2 + p^2) on (R^2, dp ^ dq). Then iota_{X_H} omega = dH gives X_H = p partial/partial q - q partial/partial p, and Hamilton's equations are dq/dt = p, dp/dt = -q. The trajectories are circles q^2 + p^2 = const — the level sets of H — traversed clockwise at unit angular speed. Energy H is constant on each circle, exactly as X_H(H) = 0 predicts.
The oscillator's X_H circulates along level sets of H; energy is conserved automatically.
Do not conflate X_H with a metric gradient: the metric gradient flows perpendicular to level sets (changing H fastest), while X_H flows parallel to them (keeping H fixed). And 'Hamiltonian' is strictly stronger than 'symplectic' for vector fields unless H^1(M; R) = 0.