Lagrangian formulation 13=
1
2
∑^3 N
α=1∑^3 N
β=1Gαβ(q 1 ,...,q 3 N) ̇qαq ̇β, (1.4.16)where
Gαβ(q 1 ,...,q 3 N) =∑N
i=1mi∂ri
∂qα·
∂ri
∂qβ(1.4.17)
is called themass metric matrixormass metric tensor. The matrixGαβis a function of
some or all of the generalized coordinates. The Lagrangian in generalized coordinates
then takes the form
L=
1
2
∑^3 N
α=1∑^3 N
β=1Gαβ(q 1 ,...,q 3 N) ̇qαq ̇β−U(r 1 (q 1 ,...,q 3 N),...,rN(q 1 ,...,q 3 N)), (1.4.18)where the potentialUis expressed as a function of the generalized coordinates through
the transformation in eqn. (1.4.14). Substitution of eqn. (1.4.18) into the Euler–
Lagrange equation, eqn. (1.4.6), gives the equations of motion foreach generalized
coordinate,qγ,γ= 1,..., 3 N:
∑^3 Nβ=1Gγβ(q 1 ,...,q 3 N) ̈qβ+∑^3 N
α=1∑^3 N
β=1[
∂Gγβ
∂qα−
1
2
∂Gαβ
∂qγ]
q ̇αq ̇β=−∂U
∂qγ. (1.4.19)
Eqn. (1.4.19) can be recast in a form that manifestly reveals the geometric structure
of the generalized coordinate space. Noting that the dyad ̇qαq ̇βis symmetric with
respect to an exchange ofαandβ, we can rewrite the second term on the left in eqn.
(1.4.19) as
∂Gγβ
∂qα
−
1
2
∂Gαβ
∂qγ=
1
2
[
∂Gγβ
∂qα+
∂Gγα
∂qβ−
∂Gαβ
∂qγ]
(1.4.20)
If we substitute eqn. (1.4.20) into eqn. (1.4.19), multiply through bythe inverse of the
mass-metric tensorG−λγ^1 , and sum overγ, we obtain an equation of motion of the form
̈qλ+∑^3 N
α=1∑^3 N
β=1Γλαβq ̇αq ̇β=−∑^3 N
γ=1G−λγ^1∂U
∂qγ(1.4.21)
where
Γλαβ=∑^3 N
γ=11
2
G−λγ^1[
∂Gγβ
∂qα+
∂Gγα
∂qβ−
∂Gαβ
∂qγ]
(1.4.22)
is known as theaffine connectionin the generalized coordinate space. On a general
manifold, an affine connection is a geometric structure that provides a way to connect
nearby tangent spaces. In the absence of forces, eqn. (1.4.21)gives the equations of
motion of geodesics (free-particle motion) in terms of the generalized coordinates.
In the remainder of this section, we will consider several examples of the use of the
Lagrangian formalism.