Why Hamilton?

By the end of the 18th century, Lagrange had shown that you can get all of Newton’s laws from a single variational principle. But sometimes it’s more convenient to swap “velocity” variables $\dot q_i$ for “momentum” variables $p_i$. That swap is done by a Legendre transform, and it leads to what we call the Hamiltonian formulation.

  1. Define the conjugate momenta
    $$ p_i = \frac{\partial L}{\partial \dot q_i}. $$
  2. Legendre transform
    $$ H(q,p,t) = \sum_i p_i\dot q_i - L(q,\dot q,t), $$ where you solve for each $\dot q_i$ in terms of $(q,p)$.

Hamilton’s Equations

Instead of one second-order equation per coordinate, you now have two first-order equations: $$ \dot q_i = \frac{\partial H}{\partial p_i}, \quad \dot p_i = -\frac{\partial H}{\partial q_i}. $$ These are Hamilton’s equations, and they’re equivalent to the Euler–Lagrange equations, but often far more flexible, especially when dealing with symmetries and conserved quantities.

Phase Space and Symplectic Geometry

  • Phase space is the $2n$-dimensional space with coordinates $(q_i,p_i)$.
  • It comes equipped with the symplectic form
    $$ \omega = \sum_i \mathrm{d}q_i \wedge \mathrm{d}p_i, $$ which encodes the “volume” preserved by the flow of Hamilton’s equations (Liouville’s theorem).
  • Modern geometric mechanics treats $(\text{phase space},\omega)$ as a smooth manifold and studies flows, symmetries, and reduction in this language.

Modern Extensions (1970s–2019)

  1. Geometric Mechanics
    Texts like Arnold’s Mathematical Methods of Classical Mechanics (1989) and Marsden & Ratiu’s Introduction to Mechanics and Symmetry (1994) recast almost everything about Hamiltonian systems in the language of differential geometry, making it easier to generalize to fluids, plasmas, and more.

  2. Field Theory
    The same extremal-action idea extends to fields. For a field $\phi(x^\mu)$, the action is $$ S[\phi] = \int \mathcal{L}\bigl(\phi,\partial_\mu\phi\bigr)\mathrm{d}^4x, $$ and one derives Hamiltonian field equations or the covariant multisymplectic formalism.

  3. Noether’s Theorem & Beyond
    Continuous symmetries of $H$ (or $\mathcal{L}$) yield conserved charges. Modern work explores generalized symmetries (e.g. higher-form symmetries) and their conserved objects.

  4. Recent Directions

    • Non-canonical brackets (e.g. Lie–Poisson systems in fluid dynamics)
    • Discrete variational integrators that preserve symplectic structure in numerical simulations
    • Resurgence and analytic continuation in semiclassical Hamiltonian systems

References

  • Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics. Springer.
  • Abraham, R., & Marsden, J.E. (1978). Foundations of Mechanics. Addison–Wesley.
  • Marsden, J.E., & Ratiu, T.S. (1994). Introduction to Mechanics and Symmetry. Springer.
  • Olver, P.J. (1993). Applications of Lie Groups to Differential Equations. Springer.