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A Classical Mechanics Concepts
A.2 Lagrange’s Equations
Lagrange showed that it is possible to formulate Newtonian mechanics in terms
of a variational principle that vastly simplifies the study of mechanical systems
in curvilinear coordinates and noninertial frames and allows a straightforward
extension to continuum mechanics. For an N particle system in three-dimensional
space, we assume there exists a function L({ ˙
q i }, {q i }, t) of generalized velocities ˙
q i
and positions q i ({ ˙
q i } denotes the set of 3N generalized velocities ( ˙
q 1 , . . . , ˙
q 3N ) and
{q i } denotes the set of 3N generalized positions (q 1 , . . . , q 3N )) such that when we
integrate L({ ˙
q i }, {q i }, t) between two points, {q i (t 1 )} and {q i (t 2 )}, in phase space,
the actual physical path is the one that extremizes the integral
S =
t 2
t 1
L({ ˙
q i }, {q i }, t)dt.
(A.2)
The function L({ ˙
q i }, {q i }, t) is called the Lagrangian and the integral S has units of
action. Extremization of the integral in Eq. (A.2) leads to the requirement that the
Lagrangian satisfy the equations
∂L
∂q i
−
d
dt
∂L
∂ ˙
q i
= 0, (i = 1, . . . , 3N).
(A.3)
Equations (A.3) are called the Lagrange equations (Goldstein 1980).
For a single particle in a potential energy field V (r), the Lagrangian is simply
L =
mv 2
2 − V (r). Note that Eqs. (A.3) are expressed directly in terms of curvilinear
coordinates. If we write down the Lagrangian in terms of curvilinear coordinates, it
is then a simple matter to obtain the equations of motion. Two important quantities
obtained from the Lagrangian are the generalized momenta,
p i =
∂L
∂ ˙
q i
,
(A.4)
and the Hamiltonian,
H =
3N
i=1
( ˙
q i p i ) − L.
(A.5)
Generalized coordinates are defined from the differential element of length ds in
real space. In Cartesian coordinates, (ds) 2 = (dx) 2 + (dy) 2 + (dz) 2 so that q 1 = x,
q 2 = y, and q 3 = z. In polar coordinates, (ds) 2 = (dr) 2 + r 2 (dθ ) 2 + (dz) 2 so that
q 1 = r, q 2 = θ , and q 3 = z. In spherical coordinates, (ds) 2 = (dr) 2 + r 2 (dθ ) 2 +
r 2 sin
2 (θ )(dφ) 2 so that q 1 = r, q 2 = θ , and q 3 = φ.
A Classical Mechanics Concepts
A.2 Lagrange’s Equations
Lagrange showed that it is possible to formulate Newtonian mechanics in terms
of a variational principle that vastly simplifies the study of mechanical systems
in curvilinear coordinates and noninertial frames and allows a straightforward
extension to continuum mechanics. For an N particle system in three-dimensional
space, we assume there exists a function L({ ˙
q i }, {q i }, t) of generalized velocities ˙
q i
and positions q i ({ ˙
q i } denotes the set of 3N generalized velocities ( ˙
q 1 , . . . , ˙
q 3N ) and
{q i } denotes the set of 3N generalized positions (q 1 , . . . , q 3N )) such that when we
integrate L({ ˙
q i }, {q i }, t) between two points, {q i (t 1 )} and {q i (t 2 )}, in phase space,
the actual physical path is the one that extremizes the integral
S =
t 2
t 1
L({ ˙
q i }, {q i }, t)dt.
(A.2)
The function L({ ˙
q i }, {q i }, t) is called the Lagrangian and the integral S has units of
action. Extremization of the integral in Eq. (A.2) leads to the requirement that the
Lagrangian satisfy the equations
∂L
∂q i
−
d
dt
∂L
∂ ˙
q i
= 0, (i = 1, . . . , 3N).
(A.3)
Equations (A.3) are called the Lagrange equations (Goldstein 1980).
For a single particle in a potential energy field V (r), the Lagrangian is simply
L =
mv 2
2 − V (r). Note that Eqs. (A.3) are expressed directly in terms of curvilinear
coordinates. If we write down the Lagrangian in terms of curvilinear coordinates, it
is then a simple matter to obtain the equations of motion. Two important quantities
obtained from the Lagrangian are the generalized momenta,
p i =
∂L
∂ ˙
q i
,
(A.4)
and the Hamiltonian,
H =
3N
i=1
( ˙
q i p i ) − L.
(A.5)
Generalized coordinates are defined from the differential element of length ds in
real space. In Cartesian coordinates, (ds) 2 = (dx) 2 + (dy) 2 + (dz) 2 so that q 1 = x,
q 2 = y, and q 3 = z. In polar coordinates, (ds) 2 = (dr) 2 + r 2 (dθ ) 2 + (dz) 2 so that
q 1 = r, q 2 = θ , and q 3 = z. In spherical coordinates, (ds) 2 = (dr) 2 + r 2 (dθ ) 2 +
r 2 sin
2 (θ )(dφ) 2 so that q 1 = r, q 2 = θ , and q 3 = φ.