Alternative formulations — from Lagrange’s elegant principle of least action to Hamilton’s geometric phase space.
Two reformulations of Newton
Analytical mechanics replaces Newton’s vector force laws with scalar energy functions, offering elegance, generality, and deep insight into the structure of physical laws.
⯞
Newtonian
Forces and acceleration. Vectors in Cartesian space. Works well for simple systems.
∂
Lagrangian
Kinetic minus potential energy. Generalized coordinates. Principle of stationary action.
⊕
Hamiltonian
Total energy as a function of position and momentum. Phase space geometry.
Why reformulate?
Newton’s laws require knowing all forces — including constraint forces. Analytical mechanics sidesteps this: constraints are automatically handled by the choice of generalized coordinates. A pendulum constrained to a rod needs only one variable θ, not the full tension vector.
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Key insight: Both the Lagrangian and Hamiltonian formulations encode the same physics, but expose different mathematical structures — the Lagrangian reveals symmetry and the principle of least action; the Hamiltonian reveals conservation laws and geometric flow in phase space.
Generalized coordinates
A system with n degrees of freedom is described by generalized coordinates q = (q₁, q₂, …, qₙ) and their time derivatives q̇ = (˙q₁, ˙q₂, …, ˙qₙ). These can be angles, distances, or any set of independent parameters that fully specify the system’s configuration.
Configuration space
The Lagrangian Formulation
Introduced by Joseph-Louis Lagrange in 1788, this formulation defines a scalar function L = T − V — the difference between kinetic and potential energy — from which all equations of motion follow.
The Lagrangian function
Definition
where T is the kinetic energy and V is the potential energy, both expressed in generalized coordinates and velocities.
Euler–Lagrange equations
The equations of motion are derived from the principle of stationary action (Hamilton’s principle): nature chooses a path between two states that makes the action integral stationary.
Action integral
Euler–Lagrange equation (for each coordinate qᵢ)
1
Write kinetic T and potential V in generalized coordinates.
2
Form L = T − V and apply the Euler–Lagrange equation for each qᵢ.
3
The result is n second-order ODEs — one per degree of freedom.
Example: Simple pendulum
For a pendulum of mass m and length ℓ, using angle θ as the single generalized coordinate:
Kinetic energy
Potential energy
Lagrangian
Equation of motion
Interactive Pendulum Simulation
Length ℓ 1.0 m
Initial angle θ₀ 45°
Damping b 0.05
Energy
KE
—
PE
—
The Hamiltonian Formulation
Developed by William Rowan Hamilton in 1833, this formulation transforms the Lagrangian via a Legendre transformation, replacing generalized velocities ˙q with generalized momenta p.
Generalized momenta
Canonical momentum
The Hamiltonian function
Legendre transform
For conservative systems where L has no explicit time dependence, H equals the total energy T + V.
Hamilton’s equations
The single 2nd-order Euler–Lagrange equation is replaced by two coupled 1st-order equations — one for each conjugate pair (qᵢ, pᵢ):
Hamilton’s canonical equations
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Phase space: The 2n-dimensional space of coordinates (q, p) is called phase space. Each state of the system is a single point, and its time evolution traces a trajectory. Hamilton’s equations define a symplectic flow — the flow preserves the phase-space volume (Liouville’s theorem).
Example: Pendulum in phase space
Hamiltonian of simple pendulum
Hamilton’s equations for pendulum
Phase Space Portrait
Each curve is a constant-energy trajectory H(θ, p) = E. Closed orbits = oscillation; open curves = full rotation.
Number of orbits 12
libration (oscillation)
rotation (full spin)
separatrix
Side-by-side comparison
Both formulations are equivalent — they describe the same physics. The choice depends on the problem’s structure and what you want to extract from it.
Lagrangian L(q, ˙q, t)
State space: configuration space (q, ˙q)
2nd-order differential equations
Natural for systems with constraints
Symmetry → cyclic coordinates → conservation
Connects to Noether’s theorem elegantly
Foundation for field theory (QFT Lagrangian)
Hamiltonian H(q, p, t)
State space: phase space (q, p)
1st-order differential equations (2n)
Natural for symplectic geometry
Liouville’s theorem: volume preserved
Direct bridge to quantum mechanics (ℋ̂)
Poisson brackets → commutators
Noether’s theorem
Emmy Noether’s 1915 theorem connects symmetry to conservation laws — one of the deepest results in physics:
Noether’s theorem (informal)
Symmetry
Time translation
→ Conservation
Energy
Symmetry
Space translation
→ Conservation
Momentum
Symmetry
Rotation
→ Conservation
Angular momentum
Poisson brackets
The Hamiltonian formalism introduces a powerful algebraic structure. For any two observables f, g:
Poisson bracket
Equation of motion for any observable
In quantum mechanics, Poisson brackets are replaced by commutators: {f, g} → [f̂, ĝ]/(iℏ).
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Historical note: Lagrange formulated his mechanics in Mécanique analytique (1788), famously boasting no figures. Hamilton published his formalism in 1833–1834. Both generalizations were crucial stepping stones to quantum mechanics, quantum field theory, and general relativity.