Classical Mechanics

1. Newtonian Formulation

Classical mechanics in its original form rests on Newton's second law, $F = ma$: given the forces acting on a particle and its position and velocity at one instant, the entire future trajectory is determined. This is the deterministic picture most people meet first — a particle traces a single, well-defined path through space, and solving the equation of motion means integrating a second-order differential equation twice, once for velocity and once for position. The formulation is powerful but coordinate-dependent and geometrically awkward once a system has constraints (a bead on a wire, a pendulum on a rod), which is precisely the gap the Lagrangian approach closes.

2. Lagrangian and the Euler–Lagrange Derivation

The Lagrangian approach reframes dynamics as an optimization problem. Define the Lagrangian $L = T - V$, kinetic energy minus potential energy, as a function of generalized coordinates $q$ and velocities $\dot q$. The claim — the principle of stationary action — is that the true path a system follows between two fixed endpoints in time is the one that makes the action

$$S = \int L(q, \dot q, t)\, dt$$

stationary against any small variation of the path. This replaces "what forces act at each instant" with "what path is singled out globally," and it works in any coordinate system, which is what makes constrained systems tractable.

Turning that variational statement into an equation of motion is a short argument in the calculus of variations: perturb the path by a small arbitrary function that vanishes at the endpoints, expand the action to first order, integrate the velocity term by parts, and demand the result vanish for every admissible perturbation. What survives is the Euler–Lagrange equation,

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right) - \frac{\partial L}{\partial q} = 0$$

one such equation per generalized coordinate. It reproduces $F = ma$ exactly when $L$ is built from ordinary kinetic and potential energy, but it generalizes cleanly to angles, constrained coordinates, and — later — fields, which is why it survives essentially unchanged into quantum field theory.

3. Hamiltonian and State Space

The Hamiltonian formulation performs a Legendre transform on the Lagrangian, trading velocities for momenta: define the conjugate momentum $p = \partial L/\partial \dot q$ and set $H(q,p) = p\dot q - L$. Rewritten this way, the single second-order Euler–Lagrange equation splits into two first-order equations, Hamilton's equations,

$$\dot q = \frac{\partial H}{\partial p}, \qquad \dot p = -\frac{\partial H}{\partial q}$$

For most physical systems $H$ is simply the total energy, kinetic plus potential, but expressed as a function of position and momentum rather than position and velocity.

The real shift is conceptual: a system's state is no longer just "where it is" but a point $(q, p)$ in phase space, the state space of classical mechanics. Every physical configuration corresponds to exactly one point, and Hamilton's equations trace out a unique flow through that space — trajectories in phase space never cross. This notion of a state, living in an abstract space and evolving under a generator called the Hamiltonian, is what carries over almost verbatim into quantum mechanics, where phase space is replaced by Hilbert space and the flow is generated by the same $H$.

4. The Poisson Bracket

Once a system is described in phase space, any two dynamical quantities $f(q,p)$ and $g(q,p)$ can be combined through the Poisson bracket,

$$\{f, g\} = \frac{\partial f}{\partial q}\frac{\partial g}{\partial p} - \frac{\partial f}{\partial p}\frac{\partial g}{\partial q}$$

summed over all coordinate pairs. It is antisymmetric, bilinear, and satisfies the Jacobi identity — an algebraic structure, not just a computational shortcut. Its physical payoff is that it governs time evolution directly: for any quantity $f$ with no explicit time dependence, $\dot f = \{f, H\}$. The bracket between a coordinate and its conjugate momentum, $\{q, p\} = 1$, is the classical shadow of what becomes the canonical commutation relation in quantum mechanics — the single most important bridge between the two formalisms.

Why it matters later: quantization is often summarized as the replacement $\{\cdot,\cdot\} \to \frac{1}{i\hbar}[\cdot,\cdot]$ — Poisson brackets become commutators, scaled by $i\hbar$. Every classical structure built from brackets has a direct quantum counterpart.

Next: First Quantization