Classical Mechanics: Lagrangian & Hamiltonian Mechanics

Classical Mechanics » Lagrangian & Hamiltonian Mechanics

Analytical mechanics reformulates Newton’s laws in terms of scalar energy functions instead of vector forces. The Lagrangian formulation derives equations of motion from a single variational principle in any convenient coordinates, eliminating constraint forces and exposing the link between symmetries and conservation laws. The Hamiltonian formulation recasts the same dynamics as a flow in phase space, which is the natural setting for statistical mechanics, chaos theory, perturbation theory, and quantization. This page covers both, ending with Hamilton-Jacobi theory and the route to quantum mechanics. The deeper geometric structure (symplectic forms, bundles, geometric phases) is treated in Geometric Formalism.

Three Formulations of One Theory

Newtonian, Lagrangian, and Hamiltonian mechanics make identical predictions for classical systems; they differ in what they take as fundamental and in which problems they make easy.

flowchart LR
    N["Newton<br/>F = dp/dt<br/>vectors in 3D space"] -->|"d'Alembert's principle<br/>(virtual work)"| La["Lagrange<br/>L(q, q̇, t)<br/>configuration space"]
    La -->|"Legendre transform<br/>p = ∂L/∂q̇"| Ha["Hamilton<br/>H(q, p, t)<br/>phase space"]
    Ha -->|"generating function<br/>that makes H = 0"| HJ["Hamilton-Jacobi<br/>PDE for S(q, t)"]
    Ha -->|"Poisson bracket → commutator/iħ"| QM["Quantum mechanics<br/>(canonical quantization)"]
    HJ -->|"ψ ≈ A e^(iS/ħ)<br/>(WKB limit)"| QM
    La -->|"sum over paths<br/>weighted by e^(iS/ħ)"| QM

Here $n$ is the number of degrees of freedom: the number of independent coordinates needed to specify the configuration after constraints are imposed. $N$ free particles have $3N$; a rigid body has 6; a double pendulum in a plane has 2.

Lagrangian Mechanics

Hamilton’s Principle

The Lagrangian of a system with kinetic energy $T$ and potential energy $V$ is

and the action of a path $q(t)$ between fixed times $t_1$ and $t_2$ is

Hamilton’s principle states that the physical path makes the action stationary: $\delta S = 0$ for every variation $\delta q(t)$ that vanishes at the endpoints. The traditional name “principle of least action” is slightly inaccurate. The action is a minimum only for sufficiently short paths; in general the true path can be a saddle point of $S$ (for example, a harmonic oscillator followed for longer than half a period).

The Euler-Lagrange Equations

Varying the path and integrating the $\delta\dot{q}$ term by parts gives

The boundary term vanishes because $\delta q_i(t_1) = \delta q_i(t_2) = 0$. For $\delta S = 0$ to hold for arbitrary independent $\delta q_i$, each bracket must vanish separately. This gives the Euler-Lagrange equations:

Two quantities appear often enough to have names:

The Lagrangian is not unique. Adding a total time derivative, $L \to L + \frac{d}{dt}F(q, t)$, changes $S$ only by boundary terms and leaves the equations of motion unchanged.

Worked Example: The Simple Pendulum

A mass $m$ on a rigid, massless rod of length $\ell$ has one degree of freedom, the angle $\theta$ from the vertical. Measuring height from the lowest point,

The two partial derivatives are $\partial L/\partial\dot\theta = m\ell^2\dot\theta$ and $\partial L/\partial\theta = -mg\ell\sin\theta$, so the Euler-Lagrange equation is

The rod tension never appears: choosing $\theta$ as the coordinate builds the constraint in. For small swings, $\sin\theta \approx \theta$ gives simple harmonic motion at $\omega_0 = \sqrt{g/\ell}$ (see Oscillations & Waves).

Constraints and Lagrange Multipliers

Constraints are classified by whether they can be written as relations among coordinates:

When it is inconvenient to eliminate coordinates, or when the constraint force itself is wanted, keep all coordinates and add one multiplier $\lambda_k$ per holonomic constraint $f_k(q, t) = 0$:

These $n$ equations plus the constraint equations determine both the motion and the $\lambda_k$. The right-hand side is the generalized constraint force. For the pendulum in Cartesian coordinates with $f = x^2 + y^2 - \ell^2$, it works out to $\lambda = -T/(2\ell)$, where $T$ is the rod tension, so the tension that the angle coordinate hid is recovered.

For nonholonomic constraints the same right-hand side is used with $\partial f_k/\partial q_i$ replaced by the coefficients $a_{ki}$ (the Lagrange-d’Alembert principle). Nonholonomic dynamics does not follow from a stationary-action principle with the constraint imposed on the varied paths; that “vakonomic” approach gives different and physically incorrect equations for rolling bodies.

Non-Conservative and Velocity-Dependent Forces

Forces that are not derived from a potential enter as generalized forces $Q_i = \sum_j \vec{F}_j \cdot \partial\vec{r}_j/\partial q_i$ on the right-hand side:

Linear viscous drag can be packaged in the Rayleigh dissipation function $\mathcal{F} = \tfrac{1}{2}\sum_i b_i \dot{q}_i^2$, with $Q_i = -\partial\mathcal{F}/\partial\dot{q}_i$.

Some forces are velocity dependent yet still fit into a Lagrangian. The most important is the Lorentz force on a charge $q$ in electromagnetic potentials $\phi$ and $\vec{A}$:

The Euler-Lagrange equations reproduce $m\ddot{\vec{r}} = q(\vec{E} + \dot{\vec{r}}\times\vec{B})$, while the canonical momentum becomes $\vec{p} = m\dot{\vec{r}} + q\vec{A}$, which is not the mechanical momentum. This distinction is the starting point for minimal coupling in quantum mechanics and gauge theory.

Noether’s Theorem

Emmy Noether’s first theorem (1918) states that every continuous symmetry of the action yields a conserved quantity. In its simplest form: if $L$ is unchanged, to first order, under $q_i \to q_i + \varepsilon\,\xi_i(q)$, then

is constant along every solution. If $L$ changes by a total derivative, $\delta L = \varepsilon\, dF/dt$, the conserved charge is $Q - F$ instead; this case covers Galilean boosts, whose conserved quantity is the center-of-mass motion. For time translation, when $\partial L/\partial t = 0$, the conserved quantity is the energy function

which equals $T + V$ when the constraints are time-independent and $V$ does not depend on velocity.

A cyclic coordinate is the simplest case: shifting $q_k$ is a symmetry, and the Noether charge is $p_k$. Local (gauge) symmetries fall under Noether’s second theorem, which gives identities among the equations of motion rather than new conserved quantities; charge conservation follows from the global part of the gauge group.

Worked Example: The Double Pendulum

For a planar double pendulum (two point masses on rigid rods), a Newtonian treatment requires both rod tensions, which change direction continuously. A Lagrangian treatment needs only the two angles. The derivation can be automated with SymPy:

import sympy as sp
from sympy.calculus.euler import euler_equations
from scipy.integrate import solve_ivp

t = sp.symbols("t")
m1, m2, l1, l2, g = sp.symbols("m1 m2 l1 l2 g", positive=True)
th1, th2 = sp.Function("theta1")(t), sp.Function("theta2")(t)

# Cartesian positions written in terms of the two generalized coordinates
x1, y1 = l1 * sp.sin(th1), -l1 * sp.cos(th1)
x2, y2 = x1 + l2 * sp.sin(th2), y1 - l2 * sp.cos(th2)

T = m1 * (x1.diff(t)**2 + y1.diff(t)**2) / 2 + m2 * (x2.diff(t)**2 + y2.diff(t)**2) / 2
V = m1 * g * y1 + m2 * g * y2
L = sp.simplify(T - V)

# Euler-Lagrange equations: no rod tensions anywhere
eqs = euler_equations(L, [th1, th2], t)
acc = sp.solve([e.lhs for e in eqs], [th1.diff(t, 2), th2.diff(t, 2)], dict=True)[0]

# Convert the symbolic accelerations into a numerical right-hand side
w1, w2 = sp.symbols("w1 w2")
subs = {th1.diff(t): w1, th2.diff(t): w2}
state, params = (th1, th2, w1, w2), (m1, m2, l1, l2, g)
a1 = sp.lambdify(state + params, acc[th1.diff(t, 2)].subs(subs))
a2 = sp.lambdify(state + params, acc[th2.diff(t, 2)].subs(subs))

p = (1.0, 1.0, 1.0, 1.0, 9.81)
def rhs(_, y):
    return [y[2], y[3], a1(*y, *p), a2(*y, *p)]

sol = solve_ivp(rhs, (0, 20), [2.0, 2.5, 0.0, 0.0], rtol=1e-10, atol=1e-10)

At large amplitudes the double pendulum is chaotic; see Chaos & Nonlinear Dynamics. For long integrations, prefer a symplectic integrator over a general-purpose one (see Computational Methods).

Small Oscillations

Expanding $L$ to second order about a stable equilibrium $q_0$ gives $L \approx \tfrac{1}{2}\dot{\eta}^{\mathsf{T}}\mathbf{M}\,\dot{\eta} - \tfrac{1}{2}\eta^{\mathsf{T}}\mathbf{K}\,\eta$ with $\eta = q - q_0$. The Euler-Lagrange equations are then linear, and the normal-mode frequencies solve $\det(\mathbf{K} - \omega^2\mathbf{M}) = 0$. This is how coupled oscillators, molecular vibrations, and lattice phonons are analyzed; see Coupled Oscillators and Normal Modes.

Hamiltonian Mechanics

The Legendre Transform

Hamilton’s formulation replaces the velocities $\dot{q}_i$ by the momenta $p_i = \partial L/\partial\dot{q}_i$ as independent variables. The function that accomplishes this change of variables is the Legendre transform of $L$ with respect to the velocities:

The transform is well defined when the Hessian $\partial^2 L/\partial\dot{q}_i\partial\dot{q}_j$ is invertible, which holds for ordinary mechanical systems. (Gauge theories violate this condition and require Dirac’s theory of constrained Hamiltonian systems.)

When $T$ is quadratic in the velocities and $V$ is velocity-independent, $H = T + V$ is the total energy. In general $H$ is conserved exactly when it has no explicit time dependence, since $dH/dt = \partial H/\partial t = -\partial L/\partial t$. For a charged particle, $H = (\vec{p} - q\vec{A})^2/2m + q\phi$.

Hamilton’s Equations

Taking the differential of $H$ and using the Euler-Lagrange equations gives Hamilton’s canonical equations:

These are $2n$ first-order equations in place of $n$ second-order ones. They also follow from a phase-space action principle, $\delta\int\left(\sum_i p_i\dot{q}_i - H\right)dt = 0$, with $q$ and $p$ varied independently.

For the pendulum, $p_\theta = m\ell^2\dot\theta$ and $H = p_\theta^2/(2m\ell^2) + mg\ell(1 - \cos\theta)$, so $\dot\theta = p_\theta/(m\ell^2)$ and $\dot{p}_\theta = -mg\ell\sin\theta$.

Phase Space and Liouville’s Theorem

The state of the system is a single point $(q, p)$ in the $2n$-dimensional phase space, and Hamilton’s equations define a velocity field on it. Solutions are the flow lines of that field. Because the system is deterministic, flow lines of a time-independent Hamiltonian never cross, and each lies on a surface of constant $H$.

The Hamiltonian flow has zero divergence in phase space:

This is Liouville’s theorem: phase-space volume is preserved by the flow. A region of initial conditions may stretch and fold, but its volume never changes. Equivalently, a phase-space density $\rho(q, p, t)$ is constant along trajectories:

Liouville’s theorem underlies equilibrium statistical mechanics (it is why the microcanonical ensemble is stationary) and rules out attractors in Hamiltonian systems, which is why dissipative and Hamiltonian chaos look so different.

Poisson Brackets

For two phase-space functions $f$ and $g$, the Poisson bracket is

It is antisymmetric, bilinear, obeys the Leibniz rule ${f, gh} = {f, g}h + g{f, h}$, and satisfies the Jacobi identity ${f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0$. The canonical coordinates satisfy

The time evolution of any observable is

so $f$ is conserved if and only if ${f, H} = 0$ (for $f$ without explicit time dependence). By the Jacobi identity, the bracket of two conserved quantities is also conserved (Poisson’s theorem). For angular momentum, ${L_x, L_y} = L_z$ and cyclic permutations, which is the classical form of the rotation algebra.

The bracket structure carries over to quantum mechanics almost unchanged, which Dirac recognized in 1925:

The correspondence is exact for $q$, $p$, and quadratic functions of them, but it cannot be extended consistently to all phase-space functions (the Groenewold-van Hove theorem). Quantization therefore involves ordering choices; deformation quantization makes the $\hbar$ corrections systematic.

Canonical Transformations

A change of phase-space variables $(q, p) \to (Q, P)$ is canonical if it preserves the form of Hamilton’s equations, with some new Hamiltonian $K(Q, P, t)$. Equivalently, it preserves the fundamental brackets, ${Q_i, P_j} = \delta_{ij}$, or the symplectic form $\omega = \sum_i dp_i \wedge dq_i$. In matrix form, the Jacobian $\mathbf{M} = \partial(Q, P)/\partial(q, p)$ satisfies $\mathbf{M}^{\mathsf{T}}\mathbf{J}\mathbf{M} = \mathbf{J}$, where $\mathbf{J} = \begin{pmatrix} 0 & \mathbf{1} \ -\mathbf{1} & 0 \end{pmatrix}$.

Canonical transformations are conveniently produced by a generating function of one old and one new variable. In every case $K = H + \partial F/\partial t$.

Examples: $F_2 = \sum_i q_i P_i$ is the identity; $F_1 = \sum_i q_i Q_i$ exchanges coordinates and momenta ($Q = p$, $P = -q$), showing that the labels “coordinate” and “momentum” have no intrinsic meaning in Hamiltonian mechanics. The time evolution of a Hamiltonian system is itself a one-parameter family of canonical transformations, which is why symplectic integrators that preserve this property have superior long-time behavior.

Action-Angle Variables

For an integrable system (one with $n$ independent conserved quantities in involution, ${F_i, F_j} = 0$), the Liouville-Arnold theorem guarantees that bounded motion lies on $n$-dimensional invariant tori and that there are canonical action-angle variables $(I, \theta)$ in which $H = H(I)$ alone:

Each $\theta_i$ increases uniformly with $2\pi$ period, and the frequencies of the motion follow from $H(I)$ without solving for the trajectory. For the harmonic oscillator the phase-space orbit is an ellipse of area $2\pi E/\omega$, so $I = E/\omega$ and $H = \omega I$.

Actions are adiabatic invariants: if a parameter of the system (a pendulum’s length, a magnetic field strength) changes slowly compared with the orbital period, $I$ stays nearly constant even though the energy does not. For a slowly shortened pendulum, $E/\omega$ is preserved. Adiabatic invariance underlies magnetic-mirror confinement of plasmas and the Bohr-Sommerfeld rule $I = n\hbar$ of the old quantum theory.

What happens to the invariant tori under a small non-integrable perturbation is the subject of KAM theory; see the KAM theory section of Chaos & Nonlinear Dynamics.

Hamilton-Jacobi Theory

The Hamilton-Jacobi Equation

The most radical simplification is a canonical transformation to variables in which the new Hamiltonian vanishes, $K = 0$, so that all new coordinates and momenta are constant. With a type-2 generating function $S(q, P, t)$, this requires $H + \partial S/\partial t = 0$ with $p_i = \partial S/\partial q_i$, which gives the Hamilton-Jacobi equation:

A complete integral $S(q, \alpha, t)$ contains $n$ independent constants $\alpha_i$, which serve as the new momenta $P_i = \alpha_i$. The new coordinates are also constant:

Solving these $n$ algebraic equations for $q_i(t; \alpha, \beta)$ gives the full motion. The solution $S$ is Hamilton’s principal function; along a physical trajectory $dS/dt = \sum_i p_i\dot{q}_i - H = L$, so $S$ is the action accumulated along the path.

Example: free particle. For $H = p^2/2m$, try $S = \alpha x - \alpha^2 t/(2m)$, which satisfies the equation. Then $\beta = \partial S/\partial\alpha = x - \alpha t/m$, so $x(t) = \beta + (\alpha/m)\,t$: uniform motion with momentum $p = \partial S/\partial x = \alpha$.

Separation of Variables

When $H$ does not depend explicitly on time, the substitution $S(q, \alpha, t) = W(q, \alpha) - Et$ gives the time-independent equation

where $W$ is Hamilton’s characteristic function. If $W$ further splits into a sum $W = \sum_i W_i(q_i)$, the problem reduces to $n$ one-dimensional quadratures. Separability in some coordinate system is closely tied to integrability; the Kepler problem, for example, separates in spherical and in parabolic coordinates, reflecting its hidden (Laplace-Runge-Lenz) symmetry.

The Route to Quantum Mechanics

Surfaces of constant $S$ in configuration space move like wavefronts, and particle trajectories are their normals ($\vec{p} = \nabla S$), just as light rays are normals to wavefronts in geometrical optics. Hamilton developed this optical-mechanical analogy in the 1830s; Schrödinger used it in 1926 to guess a wave equation whose short-wavelength limit is classical mechanics.

The connection is precise. Writing the wavefunction in polar form, $\psi = A\,e^{iS/\hbar}$, and substituting into the Schrödinger equation gives, at leading order in $\hbar$, exactly the Hamilton-Jacobi equation for $S$, plus a continuity equation for the probability density $A^2$. The first correction is the WKB approximation; the exact equation adds a “quantum potential” term proportional to $\hbar^2$.

Feynman’s path-integral formulation (1948) closes the loop with Hamilton’s principle. The quantum amplitude to go from $q_1$ to $q_2$ is a sum over all paths, each weighted by $e^{iS[q]/\hbar}$. When $S \gg \hbar$, contributions from neighboring paths cancel by rapid phase oscillation except near paths where $S$ is stationary. The classical path is selected because $\delta S = 0$ there: the principle of stationary action is the classical limit of quantum interference.

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See Also

• Newtonian Mechanics & Conservation Laws: the force-based foundation these formulations reorganize.
• Geometric Formalism: symplectic forms, Hamiltonian vector fields, and geometric phases.
• Rigid Body Dynamics: Euler angles and cyclic coordinates applied to the heavy symmetric top.
• Chaos & Nonlinear Dynamics: phase-space geometry, KAM theory, and the breakdown of integrability.
• Computational Methods: symplectic and variational integrators built on the Hamiltonian and Lagrangian structures.
• Quantum Mechanics: where Poisson brackets become commutators.