Classical Mechanics: Lagrangian & Hamiltonian Mechanics

Classical Mechanics

The principle of least action, generalized coordinates, phase space, Poisson brackets, and Hamilton-Jacobi theory.

Lagrangian Mechanics: A New Perspective

Beyond Forces: Why We Need Lagrangian Mechanics

Analyzing a bead sliding on a curved wire with Newton’s laws means constantly tracking the constraint force from the wire. Lagrange’s reformulation sidesteps this entirely: instead of forces, it focuses on energy and the path a system takes through its possible configurations, so constraint forces never need to be computed.

The Principle of Least Action

Among all possible paths between two points, a system follows the path that extremizes (usually minimizes) a quantity called the action:

Action:

Where $L = T - V$ is the Lagrangian (kinetic minus potential energy).

This single principle replaces F = ma and automatically handles constraints.

Euler-Lagrange Equations: \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0\)

These equations provide a powerful alternative to Newton’s laws. No constraint forces appear—they’re automatically accounted for by choosing appropriate coordinates.

Worked example — the simple pendulum the easy way. A mass $m$ swings on a rigid rod of length $\ell$. With Newton you would resolve the tension and gravity along the arc — bookkeeping that involves a force (the tension) you do not even care about. Lagrange asks only for the energies, written in the single natural coordinate, the angle $\theta$:

The two pieces of the Euler-Lagrange equation are

so $\frac{d}{dt}(m\ell^2\dot{\theta}) - (-mg\ell\sin\theta) = 0$, which collapses to the familiar pendulum equation:

The tension never appeared — the rigid-rod constraint was absorbed the instant we chose $\theta$ as the coordinate. For small swings, $\sin\theta \approx \theta$ gives simple harmonic motion with $\omega = \sqrt{g/\ell}$, recovering the earlier oscillations result from a completely different starting point.

Generalized Coordinates: Freedom from Cartesian Tyranny

The formulation is not restricted to $x, y, z$ — any set of coordinates that pins down the configuration will do, and the natural choice usually makes the problem simple:

• Pendulum? Use the angle θ instead of (x, y)
• Particle on a sphere? Use spherical coordinates (θ, φ)
• Double pendulum? Use two angles (θ₁, θ₂)

For a system with n degrees of freedom:

• Configuration space: Q = {q₁, q₂, …, qₙ} - all possible positions
• Generalized momentum: pᵢ = ∂L/∂q̇ᵢ - not always mass × velocity!
• Phase space: Γ = {q₁, …, qₙ, p₁, …, pₙ} - positions and momenta together

Constraints and Lagrange Multipliers

Holonomic constraints: f(q₁, …, qₙ, t) = 0

Modified Lagrangian with constraints: \(L' = L + \sum_i \lambda_i f_i(q, t)\)

Noether’s Theorem: The Deep Connection

Emmy Noether’s theorem states that every continuous symmetry of the action corresponds to a conservation law:

The Symmetry-Conservation Dictionary:

Mathematical Statement: If the action is invariant under $q_i \to q_i + \varepsilon\xi_i(q)$, then: \(Q = \sum_i \frac{\partial L}{\partial \dot{q}_i}\xi_i = \text{constant}\)

Example: Double Pendulum - Where Newton Struggles

The double pendulum illustrates the payoff. Newton’s approach would require tracking the tension forces in both rods, which constantly change direction. Lagrange needs only the energy:

def double_pendulum_lagrangian(theta1, theta2, theta1_dot, theta2_dot, 
                                m1, m2, l1, l2, g):
    """Lagrangian for double pendulum - no constraint forces needed!"""
    # Kinetic energy - just from velocities
    T = 0.5*m1*(l1*theta1_dot)**2 + 0.5*m2*((l1*theta1_dot)**2 + 
        (l2*theta2_dot)**2 + 2*l1*l2*theta1_dot*theta2_dot*
        np.cos(theta1-theta2))
    
    # Potential energy - just from positions
    V = -m1*g*l1*np.cos(theta1) - m2*g*(l1*np.cos(theta1) + 
        l2*np.cos(theta2))
    
    return T - V  # Equations of motion follow from the Euler-Lagrange equations

The Limitation of Lagrangian Mechanics

Lagrangian mechanics still thinks in terms of trajectories through configuration space. To reason more abstractly about the state of a system, the geometry of all possible motions, or the bridge to quantum mechanics, Hamilton reformulated mechanics once more — into the framework that remains central to theoretical physics today.

The Three Formulations at a Glance

All three describe the same physics — they predict identical motion — but each takes a different starting point and excels at different problems. (For the full comparison and a diagram of how they relate, see the Classical Mechanics hub.)

Hamiltonian Mechanics: The Ultimate Abstraction

From Configuration Space to Phase Space

Hamilton’s key step was to treat positions and momenta as the independent variables, rather than positions and velocities. This apparently minor change reshapes the whole perspective. In phase space:

• The evolution of a system becomes a flow
• Conservation laws become geometric properties
• The connection to quantum mechanics becomes clear

Hamilton’s Equations: Beautiful Symmetry

The Hamiltonian H typically represents total energy: \(H(q, p, t) = \sum_i p_i\dot{q}_i - L(q, \dot{q}, t)\)

Hamilton’s Canonical Equations: \(\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}\)

Phase Space: Where Physics Becomes Geometry

In phase space, a system’s state is a single point. As time evolves, this point traces a path. But here’s the beautiful part: phase space has structure.

Liouville’s Theorem: Phase space volume is preserved by time evolution \(\frac{d\rho}{dt} + \{\rho, H\} = 0\)

This means if you start with a blob of initial conditions, the blob might deform and stretch, but its volume never changes. It’s like incompressible fluid flow—a deep connection between mechanics and fluid dynamics!

Poisson Brackets: The Language of Classical Mechanics

Poisson brackets are to classical mechanics what commutators are to quantum mechanics. They encode how quantities change with time and relate to each other:

Properties:

• Antisymmetry: {f, g} = -{g, f}
• Linearity: {af + bg, h} = a{f, h} + b{g, h}
• Jacobi identity: {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0

Time evolution: df/dt = {f, H} + ∂f/∂t

This last equation is the Hamiltonian counterpart of Newton’s second law: it says every observable evolves by taking its Poisson bracket with the Hamiltonian. The fundamental brackets of the coordinates and momenta, ${q_i, p_j} = \delta_{ij}$, encode the entire structure of phase space. The parallel with quantum mechanics is exact and is the cleanest statement of the classical-quantum correspondence:

Dirac noticed this structural identity in 1925 and used it to build canonical quantization: replace Poisson brackets by commutators divided by $i\hbar$, and classical mechanics becomes quantum mechanics. The bridge is built into the formalism itself.

Canonical Transformations

Transformations preserving Hamilton’s equations: \(Q = Q(q, p, t), \quad P = P(q, p, t)\)

Generating functions:

• Type 1: $F_1(q, Q, t) \rightarrow p = \partial F_1/\partial q$, $P = -\partial F_1/\partial Q$
• Type 2: $F_2(q, P, t) \rightarrow p = \partial F_2/\partial q$, $Q = \partial F_2/\partial P$
• Type 3: $F_3(p, Q, t) \rightarrow q = -\partial F_3/\partial p$, $P = -\partial F_3/\partial Q$
• Type 4: $F_4(p, P, t) \rightarrow q = -\partial F_4/\partial p$, $Q = \partial F_4/\partial P$

Symplectic structure: Canonical transformations preserve the 2-form: \(\omega = \sum_i dp_i \wedge dq_i = \sum_i dP_i \wedge dQ_i\)

Action-Angle Variables

For integrable systems, transform to (I, θ) where:

• Action variables: Iᵢ = ∮ pᵢ dqᵢ/(2π)
• Angle variables: θᵢ with period 2π
• Hamilton’s equations: İᵢ = 0, θ̇ᵢ = ωᵢ(I) = ∂H/∂Iᵢ

Hamilton-Jacobi Theory: The Final Synthesis

Unifying Particles and Waves

The Hamilton-Jacobi equation represents the culmination of classical mechanics. It reformulates mechanics in terms of a single function S (the action), which satisfies a partial differential equation. This might seem like unnecessary abstraction, but it reveals something profound: classical mechanics has a wave-like character!

Hamilton-Jacobi Equation

The action S(q, t) satisfies: \(\frac{\partial S}{\partial t} + H\left(q, \frac{\partial S}{\partial q}, t\right) = 0\)

Complete solution: S = S(q, α, t) + const

Where α are n constants of integration.

New momenta: Pᵢ = αᵢ = constant New coordinates: Qᵢ = ∂S/∂αᵢ = βᵢ + ωᵢt

Separation of Variables

For separable systems: \(S = W(q) - Et\)

Time-independent HJ equation: \(H\left(q, \frac{\partial W}{\partial q}\right) = E\)

The Bridge to Quantum Mechanics

Here’s where Hamilton’s genius shines. The mathematical structure of classical mechanics—Poisson brackets, phase space, canonical transformations—carries over almost unchanged to quantum mechanics. Just replace:

• Poisson brackets {A, B} → Commutators [Â, B̂]/(iℏ)
• Phase space points → Wave functions
• Trajectories → Probability amplitudes

In the classical limit (ℏ → 0): \(\psi = \exp(iS/\hbar)\)

The quantum wave function reduces to classical action! This deep connection shows that classical and quantum mechanics aren’t separate theories—they’re different aspects of a unified framework.

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See Also

• Newtonian Mechanics & Conservation Laws — the force-based foundation these formulations reorganize.
• Chaos, Modern Topics & Computation — phase-space geometry, chaos, and symplectic integrators built on the Hamiltonian view.
• Quantum Mechanics — where the Poisson-bracket structure becomes the commutator algebra.