MATH4248 Weeks 10-11
Topics: Motion of a particle on a surface, calculus,
calculus of variations and Hamilton’s principle of
stationary action, kinetic energy and Riemannian
manifolds, inertial motion and geodesics, covariance,
invariance, constants of motion, rigid body motion
Objectives: To explore the mathematical meaning
of Lagrange’s equations – in particularly, its deep
connections to differential geometry and Lie groups
1
MOTION OF A PARTICLE ON A SURFACE
For a particle having mass m and constrained to move
on the surface z  h ( x , y)  0
with generalized coordinates q1  x, q 2  y
kinetic energy
2



2
2
m
T   q 1  q 2  q 1 h q 2 h  
q 2  
2
 q1


potential energy V  mgh (q , q )
1 2
Lagrangian
L  TV
2
MOTION OF A PARTICLE ON A SURFACE
Euler-Lagrange equations
d L  L  0, i  1,2
dt q i qi
generalized momenta



L  m q  q h q h  h , i  1, 2
1
2
i



q

q
q i
1
2

 q i 

and
T
2
2
h
h
 h
 h
 m ( q 1
 q 2
)(q 1
 q 2
)
q1
q 2
q1q i
q 2 q i
q i
3
MOTION OF A PARTICLE ON A SURFACE
hence
2
  h 


h

h
,
1  
 q

q1 
q1 q 2  1 


m
2 q 
 h h
 2 



h
1 

 q q

 q 2  
 1 2
2

h


 q 1 
q1 



 mA q 1q 2  mg 
h 
 2 
 q 2 
 q 2 
4
MOTION OF A PARTICLE ON A SURFACE
where


A


2

2
h
q q
1 1
h
q
1
2


2
h
q q
1 2
h
q

q q
1 2
h
q
1
 h
q q
1 2
h
q q
1 1
1
2
h

2
h
q
 h
2
2
h
q
2

 h
q q
2 2
q q
1 2
h
q
2
2
h
q
1
 h
q q
2 2
h
q
2





5
CALCULUS
Functions, their Graphs and Epigraphs
Limits and Continuity
Intermediate Values and Extreme Values
Derivatives as Linear Approximations
Integrals and The Fundamental Theorem of Calculus
Derivative of Products and Integration by Parts
Composition of Functions and the Chain Rule
Extreme and Stationary Values
Rolle’s Theorem and The Mean Value Theorem
Convexity: Geometric and Algebraic Descriptions 6
FUNCTIONS, GRAPHS, AND EPIGRAPHS
A function f : X  Y is a ‘rule’ that assigns to every x in X
(the domain) an element y in Y. The range of f, denoted by
f(X), consists of all elements in Y having the form f(x), x in X
The graph of a function f : X  Y is the subset of the
Cartesian Product X  Y that consists of all ordered pairs
(x,f(x)), x in X.
The epigraph of a function f : X R is the subset of
the Cartesian Product X  R that consists of all
ordered pairs (x,y), x in X and f ( x )  y
Problem: Prove that graph (f )  epigraph (f )
7
LIMITS
Ref = Thomas’ Calculus
Ref p.92 Let f(x) be defined on an open interval about
, except possibly at
itself. We way that f(x)
x0
x 0x approaches and write
approaches the limit L as
lim f ( x )  L
x0
x x 0
if, for every number
such that for all x ,
  0 , there exists a   0
0  | x  x 0 |    | f ( x )  L |  .
Problem: Define limits for x 0   and / or L  
8
CONTINUITY
Ref. p.125 A function y=f(x) is continuous
at an interior point c of its domain if
lim f ( x)  f (c)
x c
and is continuous at a left, right endpoint
a, b of its domain if
lim f ( x )  f (a ), lim f ( x )  f (b),
x a

x b

9
THE INTERMEDIATE VALUE THEOREM
Ref p.130 A function y=f(x) that is continuous on
a closed interval [a,b] takes on every value
between f(a) and f(b). In other words, if y 0 is any
value between f(a) and f(b), then
y 0  f ( c)
for some c in [a,b].
Problem: Interpret this in terms of the graph of f(x)
10
THE EXTREME VALUE THEOREM
Ref p. 228 If f is continuous at every point of a
closed interval I, then f assumes both an absolute
maximum value M and an absolute minimum value
m somewhere in I. That is, there are numbers
x1 and x 2 in I with f (x1 )  m, f (x 2 )  M
and m  f ( x )  M for every other x in I.
The values m and M are called absolute or global
extreme values.
Problem: What is the range of f in terms of m and M ?
11
DERIVATIVES
Ref p.147 For a function f : [a,b]  R the derivative
of the function f(x), with respect to the variable x, is
the function f ' whose value at x is
f ( x  h ) f ( x )
f ( x )  lim
h
h 0
'
provided that the limit exists. Equivalently,
f ( x  h )  f ( x )  f ( x ) h  o( h )
'
where the ratio o(h)/h 0 as h0. In a neighborhood
of x in [a,b], f(x) is the constant approximation to f
'
while the function h  f ( x ) h is a linear
approximation to f.
12
DERIVATIVES
The derivative as a linear approximation provides
the foundation for multivariable calculus
f :D  R , D  R
'
m
n
linear function f ( x ) : R  R , x  D
'
such that f ( x  h )  f ( x )  f ( x )( h )  o(h )
n
m
With respect to the standard bases on the Euclidean
spaces, h is a m x 1column vector and the derivative is
an n x m matrix valued function on D. If n = 1 and the
Euclidean dot or scalar product is considered then
f ( x )( h )  (grad f ( x ))  h
'
13
INTEGRALS AND THE
FUNDAMENTAL THEOREM OF CALCULUS
f : [a , b ]  R
Ref p. 354 Part 1. If f is continuous then
F( x ) 
x
a
'
f (u )du exists and F  f
'
Ref p. 358 Part 2. If f is continuous and F  f then
b

a
f ( x )dx 
b
F |a 
F(b)  f (a )
14
DERIVATIVE OF PRODUCTS
AND INTEGRATION BY PARTS
Ref p. 173
(uv )  uv  vu
'
'
'
Ref p. 547
b

a
b

'
b
'
u ( x ) v ( x )dx  (uv ) |a  v( x )u ( x )dx
a
Problem: Use this formula to integrate x cos x
15
COMPOSITION OF FUNCTIONS
AND THE CHAIN RULE
Ref p. 902-936
f
DR
m
g
D
 E 
 R
ER
,
n
xD
,
g f
p
D 
 R
p


g

f
(
x
)
p
Derivative of Composition R   
 R
'
m
Equals Composition of Derivatives
'
f (x)
'
g (f ( x))
R  R  R
m
n
'
'
g (f ( x))f (x)
R  R
m
p
p
16
COMPOSITION OF FUNCTIONS
AND THE CHAIN RULE
h g f
R  D  
 R , x  D
 h 1 x 1  h 1 x m 
'
'
'
  g ( y)  f ( x ) 
h (x)  



h p x 1  h p x m 
m
 g1 y1
 
g p y1



g p y n 

 g 1 y n


y=f(x)
 f1 x 1
 
f n x 1
p



x m 

 f1 x m

 f n
x
17
EXTREME AND STATIONARY VALUES
Ref p. 229 Let c be an interior point of the domain of
the function f(x). Then f ( c ) is a local maximum
value at c if and only if f ( x )  f (c) for all x
in some open interval containing c. Extreme values
are local maximum (or local minimum) values.
A stationary point of f is a point x where f ' ( x )  0
[4] p. 230 Theorem. If a function f is differentiable
at an interior point c of its domain and if f ( c ) is
an extreme value then c is a stationary point for f.
Problem: What happens for the function |x| at 0?
What happens at the ends of intervals ?
18
ROLLE’S AND MEAN VALUE THEOREMS
Ref p. 237 Rolle’s Theorem: Suppose that f is
continuous at every point of [a,b] and differentiable
at every point of (a,b). If f(a) = f(b) = 0, then there
exists c in (a,b) such that f ( c)  0.
Problem: Prove this result and then use it to prove
the following result
Ref p. 238 Mean Value Theorem: Under the
previous smoothness assumptions on f, there
exists c in (a,b) such that
f ( b )  f (a )
f (c) 
.
b a
'
19
CONVEXITY OF FUNCTIONS AND SETS
n
Definition: A subset D of R is convex if for
all a, b in D, t in [0,1] : (1  t )a  tb  D
Definition: A function f : D Ris a convex function
if D is a convex set and for all A, b in D, t in [0,1] :
f ((1  t )a  tb )  (1  t )f (a )  tf (b)
Problem: Prove that f is a convex function if and only
the epigraph of f is a convex set.
Problem: Prove that if f : [a,b]  R is continuous and
differentiable except at a finite number of points then
f is a convex function if and only if f   0.
Problem: Extend this to multivariable functions
20
CALCULUS OF VARIATIONS
Brief History: The Brachistochrone problem consists
of finding the curve in a vertical plane along which a
sliding particle will fall in the minimal time
A  (x 0 , y0 )
y
B  ( x1 , y1 )
Curve is the graph of the y=y(x) that minimizes
x1
x
dy
1 (dy / dx )
I( y) 
Fdx, F( y, ) 
dx
2g( y 0  y)
x0

2
21
CALCULUS OF VARIATIONS
This problem was solved in 1696 by Jean Bernoulli
who gave it as a challenge to other mathematicians.
It was then solved by Daniel Bernoulli, l’Hospital,
Leibniz, and Newton. By 1744 Euler developed the
modern theory of the calculus of variation, Lagrange
applied it to mechanics, and in Hamilton formulated
his Principle of Stationary Action in 1883.
Solution: the paths that connect A and B are graphs of
y  y where y( t 0 )  y( t1 )  0 and
d

F

F
I( y)  I( y  y) y 


0
dx  dy y
dx
22
CALCULUS OF VARIATIONS
This equation, obtained by Euler, can be derived
as follows. First, choose an arbitrary y
then construct the function g : R  R by
x1
dy
dy 

g(u ) 
F y  uy,  u
dx

dx
dx 
x0 

Since g has a minimum value at u = 0

d g(u ) |
d

F

F





ydx

0
u

0
dy

du
dx
 dx y 
x0 

x1 
The equation follows from Lemma on p. 57 in Arnold
since y is arbitrary and its solution is a cycloid as
23
shown in Calkin p. 63-64
STATIONARY PATHS
 

If
F  F(q, q, t ), q  (q1 ,, q f )
t1 

I(q ) 
F(q( t ), q( t ), t ) dt
t0


A path q is a stationary if for every path q



d
q( t 0 )  q( t1 )  0  I(q  uq) |  0
du
u0

iff q satisfies the Euler-Lagrange equations

F  F  d F  0,
q i q i dt q i
(evaluated at
i  1,..., f
 
(q(t ), q(t ), t ), t [t 0 , t1 ] )
24
HAMILTON’S PRINCIPLE

The actual path q
of a mechanical system is a
stationary path for the action functional defined by

S(q) 
t1
t


L(q( t ), q( t ), t ) dt
0
This is the Principle of Stationary Action
The Euler-Lagrange equations are a system of f
second-order differential equations for the fcomponent functions of the path. In most cases
the path actually minimizes the action over all
paths having the same end points.
25
GEODESICS
Consider a particle that moves along a planar curve C
v(t ), t [t 0 , t1 ]
with speed
Then

t1
t
vdt ,
0

v
t1  t 0
are the length of C, the average speed of the particle
therefore

t1

t1
2

v dt  ( v  v) dt 
t

t
1
0
t0
t0
2
is minimized by choosing v
2
 v and minimizing 
26
GEODESICS
Corollary The inertial motion (no applied force) of a
particle constrained to any surface is a constant speed
along a geodesic with respect to the line element
ds  dx  dy  dz
2
2
2
2
Proof By Hamilton’s Principle, the motion minimizes

t1

t1

t1
2
m
Ldt  Tdt 
v dt,
2 t
t0
t0
0
t1
Therefore v is constant and

t
ds 
0
ds
v
dt
t1
t
vdt
0
is minimized so the particle moves on a geodesic
27
GEODESICS
Corollary The inertial motion (no applied force) of a
system of particles with scleronomic holonomic
constraints is decribed by a geodesic with respect to
the line element
ds 
2

N

mi dx i  dyi  dzi
i 1
2
2
2

Proof By Hamilton’s Principle, the motion minimizes

t1

t1

t1
2
1
Ldt  Tdt 
v dt,
2
t0
t0
t0
ds
v
dt
28
GEODESICS
The line element is given in general coordinates
by the metric tensor G as the quadratic form
T 
f
2
ds  dq Gdq  i, j1 gijdqi dq j
The components of the metric 
tensor are

 rk  rk
N
g ij  k 1 m k

q i q j
and for a system of particles moving along a path
2


 
ds
1
T
2 dt
29
GEODESICS
Example Particle with mass m=1 on surface z=h(x,y)


rk rk
g ij 

q i q j
2
g ii  1  (h q i ) , i  1,2
g12  g12  (h q1 )(h q 2 )
Example For a surface of revolution
x  q1, y  r(q1 ) cos q 2 , z  r(q1 ) sin q 2
2
2
dr
g11  [1  (
) ], g 22  r , g12  g 21  0
dq1
30
COVARIANCE OF LAGRANGIAN
The Lagrangian L (with respect to a specified inertial
frame of reference) is a scalar valued function that is
determined by the configuration of a system – not by
the choice of generalized coordinates
Therefore, if q  q(q, t ) is any reversible
point transformation then along any path q
'
L (q, q , t )  L(q(q, t ), q (q, t ), t (q, t ))
(where q(q’,t) is the inverse point transformation)
Furthermore, the Euler-Lagrange equations are
covariant since, along the actual path
'
since these equations are equivalent
to the geometric stationarity condition
d L  L
dt q  q
'
31
INVARIANCE OF LAGRANGIAN
The Lagrangian L for a particular system is said to
be invariant under a particular transformation
q  q(q, t ) iff for any path q
'
'
'
L (q, q , t )  L (q, q , t )  L(q, q , t )
satisfies the Euler-Lagrange equations. This means
that these two Lagrangians determine the same path.
Example For the one-dimensional motion of a free
2
 2 If x  x     t then
particle L  mx
'
'
'
2
L  L  L  mx   m 2
''
''
d L  d ( m)  0  L
dt x  dt
x 
32
INVARIANCE OF LAGRANGIAN
Lemma If F(q , q
 , t ) satisfies the Euler-Lagrange
equations along every path q then the integral
I(q ) 
t
t
F(q, q , )d
0
depends only on
t 0 , q( t 0 ), t, q( t )
Proof Let Q be a path with the same ends as q. It
Suffices to show that g (0)  g (1) where
g(s)  I((1  s)q  sQ), s  [0,1]
The Fundamental Theorem of of Calculus implies that
dg
 t
t0
ds
 , )
F((1 s ) q  sQ , (1 s ) q  sQ
q
( Q  q ) d  0
33
INVARIANCE OF LAGRANGIAN
Theorem F(q , q
 , t ) satisfies the Euler-Lagrange
equations for every path q iff for some  (q , t )
d






F(q, q, t )   (q, t ) 
q
dt
q
t
Proof Fix t 0 , q( t 0 ) The lemma implies that I(q )
only depends on t , q ( t ) Hence for some  (q , t )
t
t
F(q, q , ) d  (q( t ), t )
0
The result follows again by the Fundamental Theorem
of Calculus
34
INVARIANCE OF LAGRANGIAN
Example Consider the one-dimensional motion of a
2
free particle. Then the Lagrangian L  mx
 2
is invariant under a transformation x   x  a ( t )
2
iff L''  L'  L   ma
x  ma 2
d (x, t )   x   
dt
x
t
2





 ma and
 ma 2
x
t
has the form
2
 
 

  ma 


ma 2  0
t x
x t
x
 a ( t )    t
a Galilean transformation
35
INFINITESIMAL TRANSFORMATIONS
Definition An infinitesimal transformation is a set of
point transformations q(q, , t ),   ( , )
differentiable (wrt ) and satisfying q(q,0, t )  q
    q  q  q
where q  q   and then
'
'
'
L (q, q , t )  L (q, q , t )  L(q, q , t )
 L(q, q , t )  L(q  q, q  q , t )
Lemma



L

L
d

L
  q  q    q 
q
q
dt  q

36
EMMY NOETHER’S THEOREM
Theorem If a Lagrangian L is invariant under an
infinitesimal transformation there exists a constant
of motion (or conserved quantity)
Proof The definition of invariance and the theorem on
page 34 imply that there exists a function  (q , t )
such that ''
d
L (q, q , t ) 
Then  (q, t )
implies that

(q, t ) 
dt
 (q, t )
so theorem p 36
L q   (q, t )
q
is a constant of motion
37
EMMY NOETHER’S THEOREM
Example For infinitesimal Galilean transformations
of a particle in 1-dim. x   x    t
x    t
( x, t )  mx 
therefore the quantity
L x  ( x, t )  mx (  t)  mx 
x
and hence linear momentum
and quantity
mx
mx t  mx
are conserved.
38
EMMY NOETHER’S THEOREM
Example Consider a small rotation about the z-axis
x   x  y y  y  x z  z
If the Lagrangian satisfies L'  L then it is invariant
and  ( x , t )  0 therefore (use Einstein’s rule)
L x  L y 
i
i


x i
yi
[( mi x i )(  yi )  (mi y i )( xi )]  Lz 
where Lz is the z-component of angular momentum
Hence, if the Lagrangian remains unchanged under all
rotations then angular momentum is constant in time
39
RIGID BODY MOTION
The motion of a rigid body about its center of mass is
described by a path O(t) in its configuration manifold
the rotation Lie group SO(3). It satisfies
I( t )  O( t )I(0)O( t ) inertia tensor (matrix)
angular velocity
T 
[( t )]  O( t ) O( t ),  in the body
T
OI(0)
angular momentum (in space)
Theorem The inertial motion of a rigid body about
its center of mass is descibed by Euler’s equations
T

I(0)  O OI(0)  I(0)  
40