![Page 1: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/1.jpg)
PCD_STiCM PHY101 IISER-Tirupati
1
PHY101: THE WORLD of PHYSICS I - MECHANICS
SELECT/SPECIAL TOPICS
in
CLASSICAL MECHANICS
STiCM:
PCD’s
NPTEL
course
![Page 2: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/2.jpg)
2
Alternative formulation of Mechanics via
‘Principle of Variation’.
Lagrangian/Hamiltonian formulation.
Application to SHO.
PCD-10
PCD_STiCM
Least Action Principle / Phase Space
Lagrangian / Hamiltonian formulation.
![Page 3: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/3.jpg)
3
Galileo Galilei 1564 - 1642
Galileo Newton
Lagrange
Hamilton
( , )
Linear Response.
Principle of causality.
Principle of
Variation
( , )
( , )
q q
F ma
L q q
H q p
0L d L
q dt q
,k
H Hq p
p q
PCD_STiCM
![Page 4: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/4.jpg)
The physical dimensions of a
piece of area in this space
will not be
Instead, the dimension of
area in such a space would be
4
2L
2 1L T
A two-dimensional space spanned by the two orthogonal (thus
independent) degrees of freedom.
PCD-10
PCD_STiCM
![Page 5: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/5.jpg)
5
position-momentum phase space
PCD-10
( )p t
( )q t
Dimensions?
Depends on [q], [p]
[area] : angular momentum
2 1
Accordingly, dimensions
of the position-velocity
phase will also
not always be L T
PCD_STiCM
![Page 6: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/6.jpg)
6
The mechanical state of a system
is described by
a point in phase space.
: ( , ) ( , )
: ( , ) ( , )
Coordinates q q or q p
Evolution q q or q p
PCD-10
PCD_STiCM
![Page 7: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/7.jpg)
7
Alternative principle also begins by the fact
that the mechanical system is characterized by
its position and velocity
(or equivalently by position and momentum),
…… but with a very slight difference!
PCD-10
PCD_STiCM
Newton
Principle of Causality,
Determinism
Principle of Variation
Hamilton’s Principle
Lagrange, Hamilton
![Page 8: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/8.jpg)
8
The mechanical system is determined by
a well-defined function of position and
velocity/momentum
Lagrangian and the Hamiltonian . ( , )L q q ( , )H q p
PCD-10
The alternative formulation rests is known as the
Principle of Variation.
PCD_STiCM
often referred to as the
‘Principle of Least (extremum) Action’.
![Page 9: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/9.jpg)
9
The principle of least (extremum) action: in its various
incarnations applies to all of physics.
- explains why things happen the way they do!
-Explains trajectories of mechanical systems
subject to certain initial conditions.
PCD-10
PCD_STiCM
![Page 10: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/10.jpg)
10
The principle of least action has for its precursor what is
known as Fermat's principle - explains why light takes
the path it does when it meets a boundary of a medium.
Common knowledge: when a ray of light meets the edge
of a medium, it usually does not travel along the direction
of incidence - gets reflected and refracted.
Fermat's principle explains this by stating that light travels
from one point to another along a path over which it
would need the ‘least’ time. PCD-10
PCD_STiCM
![Page 11: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/11.jpg)
11
Pierre de Fermat (1601(?)-1665) : French lawyer who pursued
mathematics as an active hobby.
Fermat's last theorem: The equation xⁿ+yⁿ=zⁿ has no non-
zero integer solutions for x,y and z for any value of n>2.
"To divide a cube into two other
cubes, a fourth power or in
general any power whatever into
two powers of the same
denomination above the second
is impossible, and I have
assuredly found an admirable
proof of this, but the margin is
too narrow to contain it."
-- Pierre de Fermat
It took about 350
years for this
theorem to be
proved (by Andrew
J. Wiles, in 1993).
PCD-10
PCD_STiCM
![Page 12: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/12.jpg)
12
Actually, the time taken by light is not necessarily a
minimum.
More correctly, the principle that we are talking about
is stated in terms of an ‘extremum’,
and even more correctly as
‘The actual ray path between two points is the one for
which the optical path length is stationary with respect
to variations of the path’.
…………………. Usually it is a minimum:
Light travels along a path that takes the least time.
PCD-10 PCD_STiCM
![Page 13: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/13.jpg)
13
Hamilton’s principle
‘principle of least (rather, extremum) action’
Hamilton's principle of least action thus has an
interesting development, beginning with Fermat's
principle about how light travels between two points, and
rich contributions made by Pierre Louis Maupertuis
(1698 -- 1759), Leonhard Paul Euler (1707 - 1783), and
Lagrange himself.
PCD_STiCM
![Page 14: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/14.jpg)
14
2
1
( , , ) .
t
t
S L q q t dtaction
Mechanical state of a system 'evolves' (along a 'world line') in such a way that
' ', is an extremum
Hamilton’s principle
‘principle of least (rather, extremum) action’
...and now, we need ‘action’,
- ‘integral’ of the ‘Lagrangian’!
PCD_STiCM
![Page 15: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/15.jpg)
PCD_STiCM 15
RIGID BODY:
Distances between every pair
of its particles is fixed.
“CONSTRAINTS”
![Page 16: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/16.jpg)
PCD_STiCM 16
dq
q q pdt
GENERALIZED GENERALIZED GENERALIZED
COORDINATE VELOCITY MOMENTUM
POSITION VELOCITY MOMENTUM
![Page 17: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/17.jpg)
x1
x2
y1 y2
P1
P2
d
X
Y
Z
P1’
P2’
P1’ P1= z1 P2’ P2= z2
17
PCD_STiCM
![Page 18: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/18.jpg)
δ
x1
x2
y1 y2
P1
P2
d d cosδ
d sinδ
d sinδ λ
d sinδ sinλ X
Y
Z
P1’
P2’
P1’ P1= z1 P2’ P2= z2
2 1
2 1
2 1
sin cos ,
sin sin ,
cos .
x x d
y y d
z z d
Two particles at fixed
distance have 5 degrees of
freedom
18
PCD_STiCM
![Page 19: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/19.jpg)
δ
x1
x2
y1 y2
P1
P2
d d cosδ
d sinδ
d sinδ λ
d sinδ sinλ X
Y
Z
P1’
P2’
P1’ P1= z1 P2’ P2= z2
2 1
2 1
2 1
sin cos ,
sin sin ,
cos .
x x d
y y d
z z d
Two particles at fixed
distance have 5 degrees of
freedom
19
PCD_STiCM
![Page 20: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/20.jpg)
δ
x1
x2
y1 y2
P1
P2
d d cosδ
d sinδ
d sinδ λ
d sinδ sinλ X
Y
Z P3
l t
P1’
P2’
P1’ P1= z1 P2’ P2= z2
P3’
P4’
P4’’
P4
Three particles at
fixed distance have 6 degrees of freedom
Rigid
body:
SIX
degrees of
freedom
20
PCD_STiCM
![Page 21: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/21.jpg)
δ
x1
x2
y1 y2
P1
P2
d d cosδ
d sinδ
d sinδ λ
d sinδ sinλ X
Y
Z
P1’
P2’
P1’ P1= z1 P2’ P2= z2
2 1
2 1
2 1
sin cos ,
sin sin ,
cos .
x x d
y y d
z z d
Two particles at fixed
distance have 5 degrees of
freedom
1 1 1, , , , :
? No!
x y z
generalized coordinates
UNIQUE
21
PCD_STiCM
![Page 22: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/22.jpg)
PCD_STiCM 22
RIGID BODY ROTATION
EULER ANGLES
………… only a very brief mention
…….
………. perhaps we can do some more
of this in tutorials?
![Page 23: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/23.jpg)
PCD_STiCM 23
XY
Z
Arbitrary rotation of the coordinate
frame from XYZ to
can be reached at by carefully choosing
exactly 3 rotations, through 3 angles
which must be chosen appropriately.
, ,
![Page 24: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/24.jpg)
PCD_STiCM 24
' cos sin 0
' sin cos 0
' 0 0 1
x x
y y
z z
Rotation about
the Z-axis
![Page 25: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/25.jpg)
PCD_STiCM 25
Rotation about the
X-axis
' 1 0 0
' 0 cos sin
' 0 sin cos
x x
y y
z z
![Page 26: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/26.jpg)
PCD_STiCM 26
Rotation about
the Y-axis
' cos 0 sin
' 0 1 0
' sin 0 cos
x x
y y
z z
![Page 27: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/27.jpg)
PCD_STiCM 27
Z-X-Z convention , ,
cos sin 0
sin cos 0
0 0 1
1 0 0
0 cos sin
0 sin cos
cos sin 0
sin cos 0
0 0 1
![Page 28: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/28.jpg)
PCD_STiCM
28
Z-X-Z
cos sin 0 1 0 0 cos sin 0
sin cos 0 0 cos sin sin cos 0
0 0 1 0 sin cos 0 0 1
c c s c s c s s c c s s
s c c c s s s c c c c s
s s s c c
cos
sin
c
s
![Page 29: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/29.jpg)
PCD_STiCM 29
XY
Z
Arbitrary rotation of the coordinate
frame from XYZ to
can be reached at by carefully choosing
exactly 3 rotations, through 3 angles
which must be chosen appropriately.
, ,
![Page 30: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/30.jpg)
PCD_STiCM 30
RIGID BODY ROTATION
EULER ANGLES
………… only a very brief mention….
………. perhaps we can do some more
of this in tutorials?
At this point, we return to
the VARIATION
PRINCIPLE
![Page 31: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/31.jpg)
31
0t 2t1t
2
1
( ) t
t
Definite f t dt integral of the function
Area under the curve
t
Integral of a function of time.
0
area of the strip
( )
in the limit
0
A f t t
t
( )f t
PCD-10
Dimensions of this ‘area’: ( )f t TPCD_STiCM
![Page 32: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/32.jpg)
32
Path Integral.
For an isolated system, dependence of L on time is not
explicit. It is implicit through dependence on position and
velocity which depend on t.
The system evolution cannot
be shown on a two-dimensional surface.
The system then evolves along a path in the ‘phase space’.
The additive property of ‘action’ as area under
the L vs. time curve remains applicable.
Thus, the dimensions of ‘action’ are equal to
dimensions of the Lagrangian multiplied by T.
We shall soon uncover what L is!
2
1
( , , ) t
t
S L q q t dtaction' ',
2
1
( ( ), ( ), ) ' ',
t
t
S L q t q t t dtaction
PCD-10
PCD_STiCM
![Page 33: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/33.jpg)
33
Consider alternative paths
(by changing the parameter )
along which the mechanical state
of the system may evolve in the
phase space : ,
q changed by q
and q changed by q
PCD-10
.
. 1 1( ), ( )q t q t
2 2( ), ( )q t q t
( )q t
( )q t
PCD_STiCM
2
1
( , )
t
t
S L q q dt
action
' '
2
1
( , )
t
t
S L q q dt
action
' '
33
![Page 34: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/34.jpg)
PCD_STiCM 34 34 PCD_STiCM
![Page 35: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/35.jpg)
35
0
SS
S
would be an extremum when the variation in is zero;
i.e.
2
1
( , , )
The mechanical system evolves in such a way that
' ', is an extremum
t
t
S L q q t dtaction
2 2
1 1
( , ) ( , ) 0
t t
t t
S L q q q q dt L q q dt
:
,
Alternative paths
q changed by q
and q changed by q
Note! ‘Force’, ‘Cause-Effect Relationship’ is NOT invoked!
PCD-10
PCD_STiCM
![Page 36: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/36.jpg)
36
( , )
dff x
dx
h hh x y x y
x y
PCD-10
PCD_STiCM
Function of 1
variable
Function of 2
variables
![Page 37: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/37.jpg)
37
0 0 0
0 0 0 0
0 0( , )
( , ) ( , )2 2lim lim
x xx y y
x xh x y h x y
h h
x x x
y
h
h: dependent
variable
x & y:
independent
variables
0 0 0
0 0 0 0
0 0( , )
( , ) ( , )2 2lim lim
y yx y x
y yh x y h x y
h h
y y y
Partial
derivatives of a
function of
more than one
variable.
PCD-10
x
PCD_STiCM
![Page 38: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/38.jpg)
38
2 2
1 1
2
1
( , , ) ( , , ) 0
i.e., 0 ( , , )
t t
t t
t
t
S L q q q q t dt L q q t dt
S L q q t dt
2
1
0
t
t
L Lq q dt
q q
We need: Integration of product of two functions PCD-10
2
1
0
t
t
L L dq q dt
q q dt
PCD_STiCM
, 0
0
d
dt
d dq q
dt dt
dq q
dt
![Page 39: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/39.jpg)
39
( ) ( ) ( ) ( )
df ( ) ( ) ( ) ( )
dx
d df dgf x g x g x f x
dx dx dx
dg df x f x g x g x
dx dx
differential and
integral of a
product of two
functions.
PCD-10
PCD_STiCM
![Page 40: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/40.jpg)
40
df
( ) ( ) ( ) ( )dx
dg df x f x g x g x
dx dx
Integrating both sides:
df( ) ( ) ( ) dx ( )
dx
dg df x dx f x g x g x dx
dx dx
2 2
2
1
1 1
df( ) ( ) ( ) | ( )
dx
x x
x
x
x x
dgf x dx f x g x g x dx
dx
2 2
1 1
2 2 1 1
df( ) ( ) ( ) ( ) ( ) ( )
dx
x x
x x
dgf x dx f x g x f x g x g x dx
dx
PCD-10
PCD_STiCM
![Page 41: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/41.jpg)
41
22 2
1 11
0
tt t
t tt
L L d Lq dt q q dt
q q dt q
2 2
1 1
0
t t
t t
L L L L dS q q dt q q dt
q q q q dt
2 2
1 1
. . 0
t t
t t
d qL Li e S q dt dt
q q dt
Integration of product of two functions
2 2
2
1
1 1
df( ) ( ) ( ) | ( )
dx
x x
x
x
x x
dgf x dx f x g x g x dx
dx
PCD-10
PCD_STiCM
![Page 42: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/42.jpg)
42
22 2
1 11
0
tt t
t tt
L L d Lq dt q q dt
q q dt q
2 2
1 1
0
t t
t t
L L L L dS q q dt q q dt
q q q q dt
2 2
1 1
. . 0
t t
t t
d qL Li e S q dt dt
q q dt
Integration of product of two functions
2 2
2
1
1 1
df( ) ( ) ( ) | ( )
dx
x x
x
x
x x
dgf x dx f x g x g x dx
dx
PCD-10 PCD_STiCM
![Page 43: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/43.jpg)
43
PCD-10
.
. 1 1( ), ( )q t q t
2 2( ), ( )q t q t
( )q t
( )q t
@t
@t
1 2
( ) at time
q t t
t t t
1 2= 0 q t q t
t1 < t < t2
22 2
1 11
0
tt t
t tt
L L d LS q dt q q dt
q q dt q
PCD_STiCM
![Page 44: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/44.jpg)
44
2
1
. . 0
t
t
L d Li e qdt
q dt q
Arbitrary variation
between the end points.
2 2
1 1
0
t t
t t
L d LS q dt q dt
q dt q
Hence, 0
'
L d L
q dt q
Lagrange s Equation
We have not, as yet,
provided a recipe to
construct the
Lagrangian!
( , ) is all the we know about it as yet!L L q q
22 2
1 11
0
tt t
t tt
L L d LS q dt q q dt
q q dt q
PCD_STiCM
![Page 45: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/45.jpg)
45
2
1 2( , , ) ( ) ( )L q q t f q f q
0 ' L d L d L L
Lagrange s Equationq dt q dt q q
Isotropy of space
L can only be quadratic function of the velocity.
2( , , ) ( )2
mL q q t q V q
T V
PCD_STiCM
![Page 46: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/46.jpg)
46
' d L L
Lagrange s Equationdt q q
i.e., : in 3D:
'
dp d PF F
dt dtNewton s II Law
2( , , ) ( )2
mL q q t q V q
T V
PCD_STiCM
, the force
, the momentum
L VF
q q
Lmq p
q
![Page 47: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/47.jpg)
47
Interpretation of L as T-V gives equivalent
correspondence with Newtonian formulation.
,
Lp
q
generalized momentum
, the force
, the momentum
L VF
q q
Lmq p
q
2( , , ) ( )2
mL q q t q V q
T V
PCD_STiCM
![Page 48: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/48.jpg)
48
( , , )
dL +
dt
L L q q t
L L Lq q
q q t
d +
dt
dL L L Lq q
dt q q t
d L Lq
dt q t
d0
dt
d
dt
L L
q q
L L
q q
Lq L
q
d L
dt t
0?L
What ift
PCD_STiCM
0L
t
![Page 49: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/49.jpg)
49
is CONSTANT
Hamiltonian
Lq L
q
H
LH q L pq L
q
2
2 2
v
1v v
2
H m L
m m V
TOTAL ENERGY
2 2 ( )H T L T T V T V
, force
, momentum
L VF
q q
Lmq p
q
0L
t
PCD_STiCM
H pq L
![Page 50: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/50.jpg)
50
1
degrees ,
N
i i
i
When there are
N of freedom
H p q L
PCD_STiCM
![Page 51: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/51.jpg)
51
:
:
:q Generalize
q Ge
d
neraliz
Vel
ed Coordi
ocit
p
y
Generalized Momen
na
tu
Lp
q
t
m
e
PCD_STiCM
![Page 52: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/52.jpg)
52
( )
Symmetry Conservation Laws
Noether
PCD_STiCM
![Page 53: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/53.jpg)
53
Lagrangian of a closed system does
not depend explicitly on time.
Hamiltonian
q is a CONSTANTL
Lq
Conservation of Energy
is thus connected with
the symmetry principle
regarding invariance with
respect to temporal
translations.
Hamiltonian: “ENERGY”
0L
t
PCD_STiCM
2 2 ( )H T L T T V T V H pq L
![Page 54: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/54.jpg)
54
since 0, this means . . is conserved.
. ., is independent of time, is a constant of motion
L d L Li e p
q dt q q
i e
In an inertial frame, Time: homogeneous;
Space is homogenous and isotropic
the condition for homogeneity of space : ( , , ) 0
. ., 0
which implies 0 where , ,
L x y z
L L Li e L x y z
x y z
Lq x y z
q
PCD_STiCM
![Page 55: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/55.jpg)
55
since 0, this means . . is conserved.
. ., is independent of time, it is a constant of motion
L d L Li e p
q dt q q
i e p
Law of conservation of momentum,
arises from the homogeneity of space.
Symmetry Conservation Laws
Momentum that is canonically conjugate to a
cyclic coordinate is conserved.
0L
q
PCD_STiCM
![Page 56: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/56.jpg)
k k k
k k k
k k k k
k k
LdH q dp dq
q
q dp p dq
56
( , )k k k k
k
H q p L q q Many degrees of freedom:
Hamiltonian (Hamilton’s Principal Function) of a system
k k k k k k
k k k kk k
L LdH p dq q dp dq dq
q q
PCD_STiCM
d0
dt
d
dt
L L
q q
L dpp
q dt
Lp
q
![Page 57: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/57.jpg)
57
k k k k
k k
dH q dp p dq
, ( , )k k
k k
k kk k
But H H p q
H Hso dH dp dq
p q
Hamilton’s equations of motion
: k k
k k
H HHence k q and p
p q
PCD_STiCM
![Page 58: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/58.jpg)
58
Hamilton’s Equations
( , )
:
k k
k k
k k
H H p q
H Hk q and p
p q
( , )k k k k
k
H q p L q q
Hamilton’s Equations of Motion :
Describe how a mechanical state of a system
characterized by (q,p) ‘evolves’ with time. PCD_STiCM
![Page 59: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/59.jpg)
59 PCD_STiCM
![Page 60: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/60.jpg)
60
[1] Subhankar Ray and J Shamanna,
On virtual displacement and virtual work in Lagrangian dynamics
Eur. J. Phys. 27 (2006) 311--329
[2] Moore, Thomas A., (2004)
`Getting the most action out of least action: A proposal'
Am. J. Phys. 72:4 p522-527
[3] Hanca, J.,Taylor, E.F. and Tulejac, S. (2004)
`Deriving Lagrange's equations using elementary calculus'
Am. J. Phys. 72:4, p510-513
[4] Hanca, J. and Taylor, E.F. (2004) `From conservation of energy to the principle of least action:
A story line'
Am. J. Phys. 72:4, p.514-521
REFERENCES
PCD_STiCM
![Page 61: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/61.jpg)
61
Applications of Lagrange’s/Hamilton’s Equations
Entire domain of Classical Mechanics
Enables emergence of ‘Conservation of Energy’
and ‘Conservation of Momentum’
on the basis of a single principle.
Symmetry Conservation Laws
Governing principle: Variational principle –
Principle of Least Action
These methods have a charm of their own and very
many applications…. PCD_STiCM
![Page 62: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/62.jpg)
62
Applications of Lagrange’s/Hamilton’s
Equations
• Constraints / Degrees of Freedom
- offers great convenience!
• ‘Action’ : dimensions
‘angular momentum’ :
: :h Max Planckfundamental quantityin Quantum Mechanics
Illustrations: use of Lagrange’s / Hamilton’s equations
to solve simple problems in Mechanics PCD_STiCM
![Page 63: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/63.jpg)
63
Manifestation of simple
phenomena in different
unrelated situations
radiation oscillators,
molecular vibrations,
atomic, molecular, solid
state, nuclear physics,
Dynamics of
spring–mass systems,
pendulum,
oscillatory electromagnetic circuits,
bio rhythms,
share market fluctuations … electrical engineering,
mechanical
engineering …
Musical instruments PCD_STiCM
![Page 64: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/64.jpg)
64
SMALL OSCILLATIONS
1581:
Observations on the
swaying chandeliers
at the Pisa cathedral.
http://www.daviddarling.info/images/Pisa_cathedral_chandelier.jp
http://roselli.org/tour/10_2000/102.htmlg
Galileo (when only 17
years old) recognized
the constancy of the
periodic time for
small oscillations. PCD_STiCM
![Page 65: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/65.jpg)
65
:
:
:
q
q
p
( , , )L L q q t
( , , )H H q p t
Generalized Coordinate
Generalized Velocity
Generalized Momentum
Lp
q
Use of Lagrange’s / Hamilton’s equations to
solve the problem of Simple Harmonic Oscillator.
PCD_STiCM
![Page 66: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/66.jpg)
66
( , , )
0 '
L L q q t Lagrangian
L d LLagrange s Equation
q dt q
( , , )
'
k k
k
k k
k k
H q p L
H H q p t Hamiltonian
H Hq p
p q
Hamilton s Equations
k: and
2nd order
differential
equation
TWO
1st order
differential
equations PCD_STiCM
![Page 67: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/67.jpg)
67
2 2
( , , )
2 2
L L q q t Lagrangian
m kL T V q q
2nd order
differential
equation
Newton’s
Lagranges’
Mass-Spring Simple Harmonic Oscillator
0
2 2 02 2
L d L
q dt q
k d mq q
dt
0
kq mq
mq kq
PCD_STiCM
![Page 68: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/68.jpg)
68
kx x
m
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Hooke.html
k
mq kq q qm
Robert Hooke (1635-1703),
(contemporary of Newton),
empirically discovered this
relation for several elastic
materials in 1678.
Linear relation between restoring force and
displacement for spring-mass system:
PCD_STiCM
![Page 69: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/69.jpg)
69
2 2: 2 2
m kLagrangian L T V q q
Lp mq
q
( , , )
( , , )
!
L L q q t
H H q p t
VERY
IMPORTANT
HAMILTONIAN approach
Note! Begin
Always with the
LAGRANGIAN.
PCD_STiCM
![Page 70: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/70.jpg)
70
2 2: 2 2
m kLagrangian L T V q q
Lp mq
q
Mass-Spring
2 2 2
2 2
m kH pq L mq q q
22
2 2
p kH q
m
PCD_STiCM
![Page 71: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/71.jpg)
71
22
2 2
p kH q
m
2
2
22k
H p pq
p m m
H kp q
q
and
( . . )'
i e f kqHamilton s Equations
TWO first order equations
PCD_STiCM
![Page 72: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/72.jpg)
72
2 2
22
:2 2
2
2
m kLagrangian L T V q q
Lp mq
q
p kH q
m
( , , )
( , , )
!
L L q q t
H H q p t
VERY
IMPORTANT
Generalized Momentum is interpreted
only as , and not a product of mass with velocity Lp
q
Be careful about how you write the Lagrangian and
the Hamiltonian for the Harmonic oscillator!
PCD_STiCM
![Page 73: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/73.jpg)
73
l : length
E: equilibrium
S: support
θ
mg
coscos
(1 cos )
h
2 2 2
( , , , )
1( ) (1 cos )
2
L L r r
L T V
L m r r mg
(1 cos )V mg
Remember this!
First: set-up the
Lagrangian
2 2
2 2
1(1 cos )
21
cos2
L m mg
L m mg mg
V mgh
r=l: constant
PCD_STiCM
![Page 74: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/74.jpg)
74
2 21cos
2L m mg mg
2
=0r
Lp
rL
p ml
Now, we can find the
generalized
momentum for each
degree of freedom.
: fixed lengthr
0 L d L
q dt q
PCD-10
PCD_STiCM
![Page 75: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/75.jpg)
75
2 21cos
2L m mg mg
0.
sin
L
rL
mgl mgl
2
=0r
Lp
rL
p ml
0 L d L
q dt q
2
2
( ) 0d
mgl mldt
mgl ml
g
l
Simple pendulum
PCD_STiCM
![Page 76: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/76.jpg)
76
0 0
0
(1)
(2) Solution:
Substitute (2) in (1)
i t i t
q q
q Ae Be
0
g
l
g
l
PCD_STiCM
![Page 77: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/77.jpg)
77
(1) Newtonian
(2) Lagrangian
g
l
0
g
l
Note! We have not used
‘force’, ‘tension in the
string’ etc. in the
Lagrangian and
Hamiltonian approach!
PCD_STiCM
![Page 78: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/78.jpg)
78
i i i
i
LH q L q p L
q
Hamilton’s equations: simple pendulum
2 2
( , , , )
1cos
2
L L r r T V
L m mg mg
2
0r
Lp ml
Lp
r
PCD_STiCM
![Page 79: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/79.jpg)
79
i
i
i i
LH q L
q
q p L
2 21cos
2rH p rp m mg mg
2
0r
Lp ml
Lp
r
2 21cos
2L m mg mg
( sin )H
mgl mgl
Hp mgl
PCD_STiCM
![Page 80: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/80.jpg)
80
2ml p mgl
g
l
Hp mgl
0
g
l
simple pendulum
0 0
0
(1)
(2) Solution:
Substitute (2) in (1)
i t i t
q q
q Ae Be
2
0r
Lp ml
Lp
r
PCD_STiCM
![Page 81: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/81.jpg)
We have NOT used ‘force’,
causality , linear-response 81
(1) Newtonian
(2) Lagrangian
(3) Hamiltonian
g
l
0
g
l
PCD_STiCM
![Page 82: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/82.jpg)
82
Lagrangian and Hamiltonian
Mechanics has very many
applications.
All problems in ‘classical
mechanics’ can be addressed using
these techniques. PCD_STiCM
![Page 83: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/83.jpg)
83
However, they do depend on the
premise that
mechanical system is characterized
by position and velocity/momentum,
simultaneously and accurately. PCD_STiCM
![Page 84: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/84.jpg)
84
Central problem in ‘Mechanics’:
How is the ‘mechanical state’ of a system
described, and how does this ‘state’ evolve
with time?
- Formulations due to Galileo/Newton,
- Lagrange and Hamilton. PCD_STiCM
![Page 85: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/85.jpg)
85
(q,p) : How do we get these?
Heisenberg’s
principle of uncertainty
New approach required !
PCD_STiCM
![Page 86: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/86.jpg)
86
‘New approach’ is not required on account of
the Heisenberg principle!
Rather,
the measurements of q and p are not compatible….
….. so how could one describe the
mechanical state of a system by (q,p) ?
PCD_STiCM
![Page 87: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/87.jpg)
87
Heisenberg principle comes into play as a
result of the fact that simultaneous
measurements of q and p do not provide
consistent accurate values on repeated
measurements. ….
….. so how could one describe the
mechanical state of a system by (q,p) ? PCD_STiCM
![Page 88: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/88.jpg)
88
| Mechanical State:
State vectors in Hilbert Space
| Measurment: C.S.C.O.
| | i Ht
Evolution of the
Mechanical
State of the system
Complete Set of Commuting Operators
Complete Set of Compatible Observables
Schrödinger Equation
Characterize? Labels?
“Good” quantum
numbers/labels
PCD_STiCM
![Page 89: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/89.jpg)
89
Galileo Galilei 1564 - 1642
Galileo Newton
Lagrange
Hamilton
( , )
Linear Response.
Principle of causality.
Principle of
Variation
( , )
( , )
q q
F ma
L q q
H q p
0L d L
q dt q
,k
H Hq p
p q
PCD_STiCM
![Page 90: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/90.jpg)
PCD_STiCM 90
![Page 91: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/91.jpg)
PCD_STiCM 91
![Page 92: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/92.jpg)
PCD_STiCM 92
![Page 93: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/93.jpg)
PCD_STiCM 93
![Page 94: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/94.jpg)
PCD_STiCM 94
![Page 95: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/95.jpg)
PCD_STiCM 95
![Page 96: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/96.jpg)
PCD_STiCM 96
![Page 97: PHY101: THE WORLD of PHYSICS I - MECHANICSlabs/amp/PHY101 Mechanics... · PHY101: THE WORLD of PHYSICS I - MECHANICS SELECT/SPECIAL TOPICS in CLASSICAL MECHANICS STiCM: PCD’s NPTEL](https://reader031.vdocument.in/reader031/viewer/2022020204/5ac440057f8b9a2b5c8ce759/html5/thumbnails/97.jpg)
PCD_STiCM 97