Hamilton's equations
Hamilton's equations are the equations of motion in phase space. Newton gave you one second-order equation per coordinate (F = m a); Lagrange gave you one second-order Euler-Lagrange equation per coordinate. Hamilton splits each of those into two first-order equations -- one for how position changes, one for how momentum changes -- and in doing so makes position and momentum stand on completely equal footing.
For a Hamiltonian H(q, p, t) with n degrees of freedom, the 2n canonical equations are q_dot_i = partial H / partial p_i and p_dot_i = - partial H / partial q_i, for i = 1..n. The first set says the velocity is the gradient of H in the momentum direction; the second says the force (rate of change of momentum) is minus the gradient of H in the coordinate direction. The near-symmetry, broken only by that one minus sign, is the fingerprint of the symplectic (canonical) structure. A coordinate that does not appear in H (a cyclic coordinate) has partial H / partial q_i = 0, so its conjugate momentum p_i is conserved -- the cleanest statement of a conservation law.
Because they are first-order, Hamilton's equations describe motion as a flow in phase space: through every point passes exactly one trajectory, and the whole system is a fluid of representative points streaming along. This is the picture that Liouville's theorem, statistical mechanics, and the entire theory of dynamical systems are built on. The total time derivative of any observable f(q, p, t) then takes the compact form df/dt = {f, H} + partial f / partial t, using the Poisson bracket.
For the harmonic oscillator H = p^2/(2m) + (1/2) m omega^2 x^2, Hamilton's equations give x_dot = partial H / partial p = p/m and p_dot = - partial H / partial x = - m omega^2 x. Differentiate the first and substitute the second: x_ddot = - omega^2 x, the familiar simple-harmonic equation, now recovered from two first-order equations.
Two first-order canonical equations reproduce the oscillator's second-order equation of motion.
The minus sign in p_dot = - partial H / partial q is not cosmetic; it is what makes phase-space flow volume-preserving (Liouville) and what distinguishes canonical dynamics from an arbitrary vector field. Swapping the roles of q and p flips its place, reflecting that (q, p) and (p, -q) are both valid canonical pairs.