Hamiltonian mechanics

Hamiltonian mechanics describes motion as flow through Phase space. The state of a system is given by coordinates $q_i$ and conjugate momenta $p_i$, and a single function $H(q,p,t)$ generates the time evolution.

Starting from Lagrangian mechanics, define

$$p_i=\frac{\partial L}{\partial {\dot{q}}_i},$$

then build the Hamiltonian by a Legendre transform:

$$H=\sum _i{p}_i{\dot{q}}_i-L.$$

Hamilton’s equations are

$${\dot{q}}_i=\frac{\partial H}{\partial p_i},{\dot{p}}_i=-\frac{\partial H}{\partial q_i}.$$

For many conservative systems, $H$ is the total Energy. More generally, treat it as the generator of time motion in phase space.

Why it matters

Hamiltonian mechanics is the clean bridge between classical mechanics and quantum mechanics. The same pattern reappears when observables become operators and Poisson brackets become commutators.

Common pitfalls

• $p_i$ is conjugate momentum, not always simply $m{\dot{q}}_i$.
• $H$ equals energy only under the right assumptions, especially no explicit time dependence and standard kinetic terms.
• Do not mix coordinate space and phase space: Hamiltonian states need both $q$ and $p$.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.