Classical mechanics can be reformulated from Newton's vector-based laws into a more powerful and elegant framework rooted in a single scalar quantity: action. Instead of forces, we consider the path a system takes through its configuration space. The principle of stationary action posits that the actual path taken by a system between two points in time, $t_1$ and $t_2$, is one for which the action is stationary (a minimum, maximum, or saddle point).

The action, denoted by $S$, is defined as the time integral of the Lagrangian, $L$. The Lagrangian itself is a scalar function that encapsulates the dynamics of the system, typically defined as the difference between the kinetic energy ($T$) and the potential energy ($V$).

To find the path that makes the action stationary, we employ the calculus of variations. We consider an arbitrary small perturbation, $\delta q(t)$, to the true path $q(t)$, such that the endpoints remain fixed ($\{\delta q(t_1) = \delta q(t_2) = 0\}$). The principle demands that the first-order change in the action, $\delta S$, must be zero for any such perturbation. This condition leads directly to the equations of motion.

The mathematical consequence of setting $\delta S = 0$ is a set of differential equations known as the Euler-Lagrange equations. For a system described by a set of generalized coordinates $q_i$, there is one Euler-Lagrange equation for each coordinate. This scalar approach is remarkably efficient, often yielding the equations of motion far more easily than a Newtonian force analysis, especially for complex systems with constraints.

While Lagrangian mechanics is a complete formulation, it yields second-order differential equations in time. The Hamiltonian formulation provides an alternative perspective that uses first-order equations and a different set of variables. The bridge between these two formalisms is the Legendre transformation, which allows us to switch from a description based on positions and velocities $(q, \dot{q})$ to one based on positions and momenta $(q, p)$.

The Hamiltonian, $H$, is constructed from the Lagrangian via the Legendre transformation. We first define the generalized (or canonical) momentum $p_i$ as the partial derivative of the Lagrangian with respect to the generalized velocity $\dot{q}_i$.

With this, the Hamiltonian is defined as:

The dynamics are then governed by Hamilton's equations, a set of 2n first-order differential equations for a system with n degrees of freedom. These equations describe the evolution of a system's state, represented as a point in phase space, a 2n-dimensional space whose axes are the generalized coordinates $q_i$ and canonical momenta $p_i$. The Hamiltonian acts as the generator of time evolution in this space.

A more abstract and powerful way to express Hamiltonian dynamics is through Poisson brackets. For any two functions $f(q, p, t)$ and $g(q, p, t)$ on phase space, their Poisson bracket is defined as:

Using this notation, the time evolution of any function $A(q, p, t)$ can be written compactly. If a quantity has a zero Poisson bracket with the Hamiltonian, it is a constant of motion, or a conserved quantity. This provides a direct test for conservation laws. Furthermore, the fundamental commutation relations in quantum mechanics are a direct analogue of the classical Poisson brackets, making this formalism an essential stepping stone to quantum theory.

This framework moves beyond simple problem-solving and provides a deep, structural understanding of classical mechanics. The principles of stationary action, the symmetries revealed by the Lagrangian, and the geometric nature of Hamiltonian phase space form the language used throughout modern theoretical physics.