Lagrangian mechanics

For two masses hung in series from springs of natural length

ℓ

, the lower equilibrium spring tension is

m_{2} g

and the upper tension is

(m_{1} + m_{2}) g

. Hence their equilibrium lengths are

L_{2} = ℓ + \frac{m _{2} g}{k _{2}}, L_{1} = ℓ + \frac{( m _{1} + m _{2} ) g}{k _{1}} .

Vertical normal modes of two equal suspended masses

For equal masses

m

and spring constants

k

, vertical displacements from equilibrium have stiffness and mass matrices

K = k (2 - 1 - 1 1), M = m I .

The two squared frequencies and displacement ratios are

ω_{\pm}^{2} = \frac{k}{2 m} (3 \pm 5), \frac{η _{2}}{η _{1}} = \frac{1 \mp 5}{2} .

Gravity shifts the equilibrium but does not enter these vertical frequencies.

Small-oscillation mass and stiffness matrices

The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix

M

and stiffness matrix

K

Generalized eigenvalue problem for small oscillations

Normal frequencies satisfy

det (K - Ω^{2} M) = 0

, and each null vector gives the fixed amplitude ratio of a mode.

Mass-matrix orthogonality of normal modes

For symmetric stiffness matrix

K

and positive-definite mass matrix

M

, eigenvectors

a_{i}, a_{j}

K a = ω^{2} M a

with distinct squared frequencies satisfy

a_{i}^{T} M a_{j} = 0.

Thus normal modes are orthogonal in the kinetic-energy inner product.

Normal modes of a double pendulum

For equal lengths and mass ratio

μ = m / M

, the squared frequencies are

ω_{0}^{2} (1 + μ \pm μ (1 + μ))

Square-root splitting of nearly degenerate frequencies

A weakly coupled degenerate pair can split by order

μ

when the small parameter enters a singular mass or stiffness matrix.

Degenerate normal modes from permutation symmetry

If the mass and stiffness matrices commute with a permutation representation, each symmetry subspace is invariant under the generalized eigenvalue problem for small oscillations. On an irreducible subspace, Schur lemma forces a commuting operator to be scalar, producing a degenerate normal mode multiplet.

Normal modes of three equal masses with a symmetric gap potential

Let three equal masses move on a circle of radius

r

, and let the potential be

U (α) + U (β) + U (γ)

with

α + β + γ = 2 π

. At equal spacing

s_{0} = 2 π /3

, angular displacements have a zero-frequency rigid-rotation mode

(1, 1, 1)

and a two-dimensional sum-zero eigenspace with

ω^{2} = \frac{3 U ^{''} ( s _{0} )}{m r ^{2}} .

The two vibrational modes are degenerate by the permutation symmetry.

Normal modes of three symmetrically coupled trapped masses

For three unit masses with unit springs between every pair and one unit spring anchoring each mass to the origin,

M = I, K = 4 I - J,

where

J

is the all-ones matrix. The collective mode

(1, 1, 1)

has angular frequency

1

, while the two-dimensional sum-zero subspace has angular frequency

2

Normal modes of two equal masses between three springs

For equal masses

m

, wall-spring constants

k_{1}, k_{2}

, and coupling-spring constant

k_{3}

, the small-oscillation mass and stiffness matrices are

M = m I, K = (k_{1} + k_{3} - k_{3} - k_{3} k_{2} + k_{3}) .

k_{1} = k (1 + ε δ)

k_{2} = k (1 - ε δ)

, and

k_{3} = k ε

, the squared normal frequencies are

ω_{\pm}^{2} = \frac{k}{m} [1 + ε (1 \pm 1 + δ^{2})] .

Conservation of energy from time-translation invariance

A Lagrangian with no explicit time dependence conserves

E = \sum_{i} \overset{q}{˙}_{i} \partial L / \partial \overset{q}{˙}_{i} - L

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Lagrangian mechanics by

Wikipedia Bot  1

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Lagrangian mechanics is a formulation of classical mechanics that uses the principle of least action to describe the motion of objects. Developed by the mathematician Joseph-Louis Lagrange in the 18th century, this approach reformulates Newtonian mechanics, providing a powerful and elegant framework for analyzing mechanical systems.

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Lagrangian mechanics by

Ciro Santilli  40 Updated 2025-07-16

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Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:

compound Atwood machine. Here, we can use the coordinates as the heights of masses relative to the axles rather than absolute heights relative to the ground
double pendulum, using two angles. The Lagrangian approach is simpler than using Newton's laws
- pendulum, use angle instead of x/y
two-body problem, use the distance between the bodies

lagrangian mechanics lectures by Michel van Biezen (2017) is a good starting point.

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector

q (t) : R \to R^{n}

that maps time to the full state:

q (t) = [q_{1} (t), q_{2} (t), q_{3} (t), ..., q_{n} (t)]

where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.

The Lagrangian is a function

R^{n} \times R^{n} \times R \to R

that maps:

L (q (t), \dot{q} (t), t)