STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 02:PCD_STiCM Unit 1 Equations of Motion (i) 1 Unit 1: Equations of Motion
Equations of Motion. Principle of Causality and Newton’s I & II Laws. Interpretation of Newton’s 3rd Law as ‘conservation of momentum’ and its determination from translational symmetry.
Alternative formulation of Mechanics via ‘Principle of Variation’. Determination of Physical Laws from Symmetry Principles, Symmetry and Conservation Laws.
Lagrangian/Hamiltonian formulation. Application to SHO. PCD_STiCM 2 PCD-10 Learning goals:
‘Mechanical system’ is described by (position, velocity) or (position, momentum) or some well defined function thereof.
How is an ‘inertial frame of reference’ identified? What is the meaning of ‘equilibrium’? What causes departure from equilibrium? Can we ‘derive’ I law from the II by putting F0 in F ma . PCD_STiCM 3 PCD-10 ……. Learning goals:
Learn about the ‘principle of variation’ and how it provides us an alternative and more powerful approach to solve mechanical problems.
Introduction to Lagrangian and Hamiltonian methods which illustrate this relationship and apply the technique to solve the problem of the simple harmonic oscillator.
PCD_STiCM 4 PCD-10 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. Why ‘position’ and ‘velocity’ are both needed to specify the mechanical state of a system?
They are independent parameters that specify the ‘state’. The mechanical state of a system is characterized by its position and velocity, ()qq, or, position and momentum, () qp,
Or, equivalently by their well-defined functions: L(,): q q Lagrangian H(,): q p HamiltonianPCD_STiCM 5 PCD-10 A two-dimensional space spanned by the two orthogonal (thus independent) degrees of freedom.
The physical dimensions of a piece of area in this space 2 will not be L
Instead, the dimension of area in such a space would be LT21 PCD_STiCM 6
PCD-10 position-momentum phase space
Dimensions? Depends on [q], [p] Accordingly, dimensions of the position-velocity phase will also pt() not always be LT21 [area] : angular momentum
qt() PCD_STiCM 7
PCD-10 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 )
What is meant by ‘Equation of Motion’ ?
- Rigorous mathematical relationship between position, velocity and acceleration. PCD_STiCM 8
PCD-10 Galileo’s experiments that led him to the law of inertia.
Galileo Galilee 1564 - 1642
PCD_STiCM 9
PCD-10 What is ‘equilibrium’? What causes departure from ‘equilibrium’? Galileo Galilei Isaac Newton 1564 - 1642 (1642-1727)
Causality & Determinism
I Law II Law
F ma Effect is proportional to the C ause . Linear Response. Principle of causality. PCD_STiCM 10
PCD-10 F ma Effect is proportional to the C ause . Linear Response. Principle of causality.
Now, we already need calculus!
PCD_STiCM 11
PCD-10 From Carl Sagan’s ‘Cosmos’: “Like Kepler, he (Newton) was not immune to the superstitions of his day… in 1663, at his age of twenty, he purchased a book on astrology, …. he read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments, So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.”
PCD_STiCM 12
PCD-10 fx
Differential of a function. Derivative of a function Slope of the curve. x It is a quantitative measure of how sensitively the function f(x) responds to changes in the independent variable x .
The sensitivity may change from point to point and hence the derivative of a function must be determined at each value of x in the domain of x. PCD_STiCM 13
PCD-10 xx f()() x f x df f 00 lim lim 22 xx00 fx dx xx x x 00
f fx 0
df x fx dx x0
x
Tangent to the curve. x0 Dimensions of the derivativePCD_STiCM of the function: [f][x]-1 14 PCD-10 Functions of many variables: eg. height above a flat horizontal surface of a handkerchief that is stretched out in a warped surface, …… y
h One must then define h: dependent variable ‘Partial derivatives’ x & y: of a function independent of more than variables one variable. x Or, the temperature on a flat surface that has various heat sources spread under it – such as a tiny hot filament here, a hotter there, a tiny ice cube here, a tiny beaker of liquid heliumPCD_STiCM there, etc. 15 PCD-10 xx h(,)(,) x y h x y hh 022 0 0 0 lim lim x xx00 x x (,)x0 y 0y y 0 h
h: dependent variable Partial x & y: derivatives of a independent function of variables more than one variable. x yy h(,)(,) x y h x y hh 0 0 0 0 lim22 lim yy00 y (,)x y y y x 0 0PCD_STiCM 016
PCD-10 Time-Derivatives of t t21 t position & rt() 1 r velocity. Y r I v lim r()() t r t velocity 21 t0 t OI XI r()() t21 r t ZI rt()2 v lim t0 t
‘Equation of Motion’ Rigorous relationship between v a lim acceleration position, t0 t velocity v(tt ) v( ) and acceleration. a lim 21 PCD_STiCM t0 t 17 PCD-10 I Law: Law of Inertia ----- COUNTER-INTUITIVE
What is ‘equilibrium’? F ma Relative to whom? Effect a is proportional - frame of reference to the Cause F . Equilibrium means Proportionality: Mass/Inertia. ‘state of rest’, or of uniform motion along a Linear Response. straight line. Principle of causality/determinism. Equilibrium sustains itself, needs no cause; Galileo; Newton determined entirely by
initial conditions. PCD_STiCM 18
PCD-10 Force: Physical agency that changes the state of equilibrium of the object on which it acts. d2 r dv d ( m v) dp m m ma F . dt2 dt dt dt
dd F r v and v tt dt dt m
Direction of velocity and acceleration both reverse.
System’s trajectory would be only reversed along essentially the same path.
Newton’s laws are therefore symmetric under time-reversal.
PCD_STiCM 19
PCD-10 Newton’s laws: ‘fundamental’ principles – laws of nature. Did we derive Newton’s laws from anything?
We got the first law from Galileo’s brilliant experiments.
We got the second law from Newton’s explanation of departure from equilibrium in terms of the linear-response cause-effect relationship.
No violation of these predictions was ever found.
- especially elevated status: ‘laws of nature’; - universal in character since no mechanical phenomenon seemed to be in its range of applicability. PCD_STiCM 20
PCD-10 We have introduced the first two laws of Newton as fundamental principles. Newton’s III law makes a qualitative and quantitative statement about each pair of interacting objects, which exert a mechanical force on each other.
FF12 21 In Newtonian scheme of mechanics, this is introduced as a ‘fundamental’ principle
–i.e., asPCD_STiCM a law of nature. 21
PCD-10 Newton’s III law : ‘Action and Reaction are Equal and Opposite’
FF12 21 d p d p 12 dt dt d pp0 dt 12 Newton' s III Law We have obtained a as statement of conservation principle from conservation of ‘law of nature’ linear momentum
PCD_STiCM 22
PCD-10 Are the conservation principles consequences of the laws of nature? Or, are the laws of nature the consequences of the symmetry principles that govern them?
Until Einstein's special theory of relativity, it was believed that conservation principles are the result of the laws of nature.
Since Einstein's work, however, physicists began to analyze the conservation principles as consequences of certain underlying symmetry considerations, enabling the laws of nature to be revealed from this analysis.
PCD_STiCM 23
PCD-10 Instead of introducing Newton’s III law as a fundamental principle, we shall now deduce it from symmetry / invariance.
This approach places SYMMETRY ahead of LAWS OF NATURE. It is this approach that is of greatest value to contemporary physics. This approach has its origins in the works of Albert Einstein, Emmily Noether and Eugene Wigner.
PCD_STiCM 24 (1879 – 1955) (1882 – 1935) (1902 – 1995) PCD-10 STiCM
Select / Special Topics in Classical Mechanics
NEXT CLASS: STiCM Lecture 03: Unit 1 Equations of Motion (ii)
PCD_STiCM 25
PCD-10 STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 03:PCD_STiCM Unit 1 Equations of Motion (ii) 26
PCD-L03 Newton’s III law : ‘Action and Reaction are Equal and Opposite’
FF12 21 d p d p 12 dt dt d pp0 dt 12 Newton' s III Law We have obtained a as a statement of conservation principle from conservation of ‘law of nature’ linear momentum
PCD_STiCM 27
PCD-L03 Are the conservation principles consequences of the laws of nature? Or, are the laws of nature the consequences of the symmetry principles that govern them?
Until Einstein's special theory of relativity, it was believed that conservation principles are the result of the laws of nature.
Since Einstein's work, however, physicists began to analyze the conservation principles as consequences of certain underlying symmetry considerations, enabling the laws of nature to be revealed from this analysis.
PCD_STiCM 28
PCD-L03 Before we proceed, we remind ourselves of another illustration of the connection between symmetry and conservation law.
Examine the ANGULAR MOMENTUM l r p of a system subjected to a central force.
dl d SYMMETRY r p r F, F F eˆr dt dt 0 CONSERVED dr dp since pF 0 and . l: constant QUANTITY dt dt
PCD_STiCM 29
PCD-L03 The connection between ‘symmetry’ and ‘conservation law’ is so intimate, that we can actually derive Newton’s III law using ‘symmetry’ .
Begin with a symmetry principle: translational invariance in homogenous space.
Consider a system of N particles in a medium that is homogenous.
A displacement of the entire N-particle system through S in this medium would result in a new configuration that would find itself in an environment that is completely indistinguishable from the previous one. This invariance of the environment of the entire N particle system following a translational displacement is
a result of translational symmetryPCD_STiCM in homogenous space. 30
PCD-L03 FOUR essential considerations: 1. Each particle of the system is under the influence only of all the remaining particles; the system is isolated: no external forces act on any of its particles.
2 The entire N-particle system is deemed to have undergone simultaneous identical displacement; all inter-particle separations and relative orientations remain invariant.
3 Entire medium: essentially homogeneous; - spanning the entire system both before and after the displacement.
4. The displacement the entire system under consideration
is deemed to take place PCD_STiCMat a certain instant of time. 31
PCD-L03 The implication of these FOUR considerations:
“Displacement considered is only a virtual displacement.”
- only a mental thought process;
- real physical displacements would require a certain time interval over which the displacements would occur
- virtual displacements can be thought of to occur at an instant of time and subject to the specific four features mentioned.
PCD_STiCM 32
PCD-L03 Displacement being virtual,
it is redundant to ask
“what agency has caused it”
PCD_STiCM 33
PCD-L03 Now, the internal forces do no work in this virtual displacement.
Therefore ‘work done’ by the internal forces in the ‘virtual displacement’ must be zero.
This work done (rather not done) is called ‘virtual work’. NNN 0,W Fk s F ik s k1 k 1 i 1, i k th th where Fik : force on k particle by the i , and
th Fkk force on the particle due to the remaining N -1 particles.PCD_STiCM 34 PCD-L03 NNN 0 W Fk s F ik s k1 k 1 i 1, i k The mathematical techniques: Jean le Rond d'Alembert (1717 – 1783)
Under what conditions can the above relation hold for an ARBITRARY displacement s ?
NNN dpk d dP 0 Fpk k k1 k 1dt dt k 1 dt Newton’s I,II laws used; not the III. PCD_STiCM 35
PCD-L03 NNNdp d dP The relation k 0 Fpk k k1 k 1dt dt k 1 dt is obtained from the properties of translational symmetry in homogenous space. Amazing! For just two particles:
dP - since it suggests a path to 0, dt discover the laws of physics d p d p ie. ., 21 , by exploiting the connection dt dt between symmetry and which gives FF12 21 , the III law of Newton . conservation laws! PCD_STiCM 36
PCD-10 Newton’s III law need not be introduced as a fundamental principle/law; we deduced it from symmetry / invariance.
SYMMETRY placed ahead of LAWS OF NATURE.
Albert Einstein, Emmily Noether and Eugene Wigner.
PCD_STiCM 37 (1879 – 1955) (1882 – 1935) PCD-L03 (1902 – 1995)
Emilly Noether: Symmetry Conservation Laws
Eugene Wigner's profound impact on physics: symmetry considerations using `group theory' resulted in a change in the very perception of just what is most fundamental.
`symmetry' : the most fundamental entity whose form would govern the physical laws.
PCD_STiCM 38
PCD-10 The connection between SYMMETRY and CONSERVATION PRINCIPLES brought out in the previous example, becomes even more transparent in an alternative scheme of MECHANICS.
While Newtonian scheme rests on the principle of causality (effect is linearly proportional to the cause), this alternative principle does not invoke the notion of ‘force’ as the ‘cause’ that must be invoked to explain a system’s evolution.
PCD_STiCM 39
PCD-10 This 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_STiCM 40
PCD-10 The mechanical system is determined by a well-defined function of position and velocity/momentum.
The functions that are employed are the Lagrangian L (,) q q and the Hamiltonian H (,) q p .
The primary principle on which this alternative formulation rests is known as the Principle of Variation. PCD_STiCM 41
PCD-10 We shall now introduce the ‘Principle of Variation’,
often referred to as the ‘Principle of Least (extremum) Action’.
PCD_STiCM 42
PCD-10 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_STiCM 43
PCD-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_STiCM 44
PCD-10 Pierre de Fermat (1601(?)-1665) : French lawyer who pursued mathematics as an active hobby.
Best known for what has come to be known as Fermat's last theorem, namely that 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 It took about 350 I have assuredly found an years for this admirable proof of this, but the theorem to be margin is too narrow to contain it." proved (by Andrew J. Wiles, in 1993). -- Pierre de FermatPCD_STiCM 45
PCD-10 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_STiCM 46
PCD-10 STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 04:PCD_STiCM Unit 1 Equations of Motion (iii) 47
PCD-L04 We shall now introduce the ‘Principle of Variation’,
often referred to as the ‘Principle of Least (extremum) Action’.
PCD_STiCM 48
PCD-10 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.
Hamilton’s principle
‘principle of least (rather,PCD_STiCM extremum) action’ 49 Hamilton’s principle
‘principle of least (rather, extremum) action’
Mechanical state of a system 'evolves' (along a 'world line') in such a way that t2 'action ', S L(,,). q q t dt is an extremum t1
...and now, we need ‘action’, - ‘integral’ of the ‘Lagrangian’!
PCD_STiCM 50 Integral of a function of time. t2 Definite integral of the function f() t dt t1 ft() Area under the curve
area of the strip
A f() t0 t in the limit t 0
t t0 t 1 2 t Dimensions of this ‘area’: f() t T PCD_STiCM 51
PCD-10 t2 Path Integral. 'action ', S L(,,) q q t dt t1 Dependence of L on time is not explicit. It is implicit through dependence on position and velocity which depend on t. t2 'action ', S L( q ( t ), q ( t ), t ) dt t1 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 discoverPCD_STiCM what L is! 52 PCD-10 qt() q( t ), q ( t ) . 11
q( t22 ), q ( t ) .
Consider alternative paths along which the mechanical state qt() of the system may evolve in the phase space : q changed by q ,
and q changed by PCD_STiCM q 53
PCD-10 Alternative paths : q changed by q , and q changed by q The mechanical system evolves in such a way that t2 'action ', S L(,,) q q t dt is an extremum t1 Note! ‘Force’, ‘Cause-Effect Relationship’ is NOT invoked! S would be an extremum when the variation in S is zero; i.e. S 0
tt22 S Lq( qq , qtdt , ) Lqqtdt ( , , ) 0 tt 11PCD_STiCM 54
PCD-10 df fx dx
hh h(,) x y x y xy
PCD_STiCM 55
PCD-10 xx h(,)(,) x y h x y hh 022 0 0 0 lim lim x xx00 x x (,)x0 y 0y y 0 h
h: dependent variable Partial x & y: derivatives of a independent function of variables more than one variable. x yy h(,)(,) x y h x y hh 0 0 0 0 lim22 lim yy00 y (,)x y y y x 0 0PCD_STiCM 056
PCD-10 tt22 S Lq( qq , qtdt , ) Lqqtdt ( , , ) 0 tt11 t2 i.e., 0S L ( q , q , t ) dt t1 t2 LL 0 q q dt qq t1 t2 L L d 0 q q dt q q dt t1
We need: Integration ofPCD_STiCM product of two functions 57
PCD-10 differential and integral of a product of two functions. d df dg f()()()() x g x g x f x dx dx dx
dg d df f ( x ) f ( x ) g ( x ) - g ( x ) dx dx dx
PCD_STiCM 58
PCD-10 dg d df f()()()-() x f x g x g x dx dx dx
Integrating both sides: dg d df fx() dx fxgx ()()dx- gxdx () dx dx dx
xx22 dg x2 df fx( ) dxfxgx ( ) ( ) |x gxdx ( ) dx 1 dx xx11
xx 22dg df fx()()()()()() dxfxgx2 2 fxgx 1 1 gxdx dx dx xx11
PCD_STiCM 59
PCD-10 tt22L L L L d 0 S q q dt q q dt q q q q dt tt11
tt22LLdq i. e . 0 S q dt dt q q dt tt11
Integration of product of two functions
xx22 dg x2 df fx( ) dxfxgx ( ) ( ) |x gxdx ( ) dx 1 dx xx11
t tt22L L 2 d L 0q dt q q dt ttq q t dt q 11PCD_STiCM1 60
PCD-10 tt22L L L L d 0 S q q dt q q dt q q q q dt tt11
tt22LLdq i. e . 0 S q dt dt q q dt tt11
Integration of product of two functions
xx22 dg x2 df fx( ) dxfxgx ( ) ( ) |x gxdx ( ) dx 1 dx xx11
t tt22L L 2 d L 0q dt q q dt ttq q t dt q 11PCD_STiCM1 61
PCD-10 q( t ) at time t qt() @t q( t11 ), q ( t ) t12 t t . @t
q( t22 ), q ( t ) . t1 < t < t2
q t12= q t 0
qt()
t tt22L L 2 d L 0S q dt q q dt ttq q t dt q 11PCD_STiCM1 62
PCD-10 STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 05:PCD_STiCM Unit 1 Equations of Motion (iv) 63
PCD-L05 Mechanical state of a system 'evolves' (along a 'world line' ) in such a way that t2 'action ', S L(,,). q q t dt is an extremum t1
tt22 S Lq( qq , qtdt , ) Lqqtdt ( , , ) 0 tt11 t2 i.e., 0S L ( q , q , t ) dt t1
t tt22L L 2 d L 0S q dt q q dt ttq q t dt q 11PCD_STiCM 1 64 q( t ) at time t qt() @t q( t11 ), q ( t ) t12 t t . @t
q( t22 ), q ( t ) . t1 < t < t2
q t12= q t 0 qt()
t tt22L L 2 d L 0S q dt q q dt ttq q t dt q 11PCD_STiCM1 65 t tt22L L 2 d L 0S q dt q q dt q q dt q tt11 t1 tt22L d L 0S q dt q dt q dt q tt11 t2 L d L Arbitrary variation i. e . 0 qdt q dt q t1 between the end points. L d L We have not, as yet, Hence, 0 provided a recipe to q dt q Lagrange' s Equation construct the Lagrangian! L L( q , q ) is all the PCD_STiCMwe know about it as yet! 66 L d L d L L 0 Lagrange ' s Equation q dt q dt q q Homogeneity & Isotropy of space
L can only be quadratic function of the velocity. 2 L(,,)()() q q t f12 q f q m L(,,)() q q t q2 V q 2
PCD_STiCMTV 67 d L L Lagrange' s Equation dt q q m L(,,)() q q t q2 V q 2 LV TV - F, the force qq L mq p, the momentum q dp d P i.e., FF : in 3D: dt dt Newton' s II Law PCD_STiCM 68 m L L(,,)() q q t q2 V q p , 2 q TV generalized momentum
LV F, the force qq L mq p, the momentum q
Interpretation of L as T-V gives equivalent correspondence with Newtonian formulation. PCD_STiCM 69 L L(,,) q q t LL d 0 qq dt dL LLL LL d qq + dt q q t qq dt
dLd L L L qq + dtdt q q t d L L q dt q t dLL qL L dtq t What if 0? PCD_STiCM t 70 L L qL - is CONSTANT 0 q t
Hamiltonian's Principal Function L LV H q - L pq - L F, force q qq L mq p, momentum H mv2 L q
221 mvv m V 2 HTLTTVTV2 - 2 -( - ) TOTAL ENERGY PCD_STiCM 71 When there are
N degrees of freedom ,
N H pii q L i1
PCD_STiCM 72 L p q q: Generalized Coordinate q: Generalized Velocity
p : Generalized Momentum
PCD_STiCM 73 Symmetry Conservation Laws
()Noether
PCD_STiCM 74 L Lagrangian of a closed system does 0 not depend explicitly on time. t L Conservation of Energy q - L is a CONSTANT is thus connected with q the symmetry principle Hamiltonian: “ENERGY” regarding invariance with respect to temporal translations.
Hamiltonian / Hamilton’s Principal Function
PCD_STiCM 75 In an inertial frame, Time: homogeneous; Space is homogenous and isotropic
the condition for homogeneity of space : L ( x , y , z ) 0 LLL i. e ., L x y z 0 x y z L which implies 0 where q x , y , z q
L d L L since 0, this means i . e . p is conserved. q dt q q ie. ., is independent of time, is a constant of motion
PCD_STiCM 76 L 0 q
L d L L since 0, this means i . e . p is conserved. q dt q q i. e ., p is independent of time, it is a constant of motion
Law of conservation of momentum, arises from the homogeneity of space. Symmetry Conservation Laws
Momentum that is canonically conjugate to a cyclic coordinate is PCD_STiCMconserved. 77 Hamiltonian (Hamilton’s Principal Function) of a system
Many degrees of freedom: H qk p k L(,) q k q k k
LL dH pk dq k q k dp k dq k dq k k k kqqkk k L dH qk dp k dq k kkqk
qk dp k p k dq k kkPCD_STiCM 78 dH qk dp k p k dq k kk
But,(,) H H pkk q
HH so dH dpkk dq kkpqkk HH Hence k: qkk and p pqkk Hamilton’s equations of motion PCD_STiCM 79 Hamilton’s Equations H qk p k L(,) q k q k k
H H(,) pkk q
HH k: qkk and p pqkk
Hamilton’s Equations of Motion : Describe how a mechanical state of a system characterized by (q,p) ‘evolves’ with time. PCD_STiCM 80 PCD_STiCM 81 REFERENCES [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
NEXT CLASS: STiCM LecturePCD_STiCM 06: Unit 1 Equations of Motion 82(v) STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 06:PCD_STiCM Unit 1 Equations of Motion (v) 83 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 84 Applications of Lagrange’s/Hamilton’s Equations
• Constraints / Degrees of Freedom - offers great convenience!
• ‘Action’ : dimensions ‘angular momentum’ : h:: Max Planck fundamental quantity in Quantum Mechanics
Illustrations: use of Lagrange’s / Hamilton’s equations to solve simplePCD_STiCM problems in Mechanics 85 Manifestation of simple phenomena in different unrelated situations Dynamics of spring–mass systems, pendulum, oscillatory electromagnetic circuits, bio rhythms, share market fluctuations … radiation oscillators, electrical engineering, molecular vibrations, mechanical atomic, molecular, solid engineering … state, nuclear physics, PCD_STiCM Musical instruments86 SMALL OSCILLATIONS
1581: Observations on the swaying chandeliers at the Pisa cathedral.
Galileo (when only 17 years old) recognized the constancy of the periodic time for small oscillations. PCD_STiCMhttp://www.daviddarling.info/images/Pisa_cathedral_chandelier.jp 87 http://roselli.org/tour/10_2000/102.htmlg Use of Lagrange’s / Hamilton’s equations to solve the problem of Simple Harmonic Oscillator. q : Generalized Coordinate
q : Generalized Velocity
p : L p Generalized Momentum q L L(,,) q q t H H(,,) q p t
PCD_STiCM 88 L L( q , q , t ) Lagrangian 2nd order
L d L 0 Lagrange ' s Equation differential q dt q equation
H qkk p L k
H H(,,) q p t Hamiltonian TWO
1st order HH k: qpkk and pqkkdifferential equations Hamilton' s EquationsPCD_STiCM 89 Mass-Spring Simple Harmonic Oscillator
L L( q , q , t ) Lagrangian L d L 0 mk L T V q22 q q dt q 22
k d m kq mq 0 2qq 2 0 22dt
mq kq 2nd order
Newton’s differential Lagranges’ PCD_STiCMequation 90 k mq kq q q m Linear relation between restoring force and displacement for spring-mass system:
Robert Hooke (1635-1703), k xx (contemporary of Newton), m empirically discovered this relation for several elastic materials in 1678.
PCD_STiCM 91 http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Hooke.html HAMILTONIAN approach L L(,,) q q t Note! Begin H H(,,) q p t Always with the LAGRANGIAN. VERY IMPORTANT !
mk Lagrangian: L T V q22 q 22 L p mq q
PCD_STiCM 92 Mass-Spring mk Lagrangian: L T V q22 q 22 L p mq q mk H pq L mq2 q 2 q 2 22
pk2 Hq2 22m PCD_STiCM 93 pk2 Hq2 22m
H2 p p q p2 m m
Hk and pq 2 qk 2 (..)i e f kq Hamilton' s Equations
TWOPCD_STiCM first order equations 94 Be careful about how you write the Lagrangian and the Hamiltonian for the Harmonic oscillator! mk L L(,,) q q t Lagrangian: L T V q22 q 22
L p mq q H H(,,) q p t
pk2 Hq2 VERY 2m 2 IMPORTANT ! Generalized Momentum is interpreted
L only as p , and not a product of mass with velocity q PCD_STiCM 95 S: supporth cos θ cos (1 cos ) l : length V mgh V mg (1 cos ) mg E: equilibrium Remember this! L L(,,,) r r LTV First: set-up the 1 L m( r2 r 2 2 ) mg (1 cos ) Lagrangian 2 1 L m22 mg (1 cos ) r=l: constant 2 1 22 LPCD_STiCM m mg mg cos 96 2 1 L m22 mg mg cos 2 r : fixed length Now, we can find the generalized momentum for each degree of freedom. L p =0 r r L d L L 2 0 p ml q dt q PCD_STiCM 97
PCD-10 Simple pendulum 1 L m22 mg mg cos 2 L 0. L d L r 0 L q dt q mglsin mgl d L mgl ( ml 2 ) 0 pr =0 dt r mgl ml 2 L 2 g p ml PCD_STiCM l 98 g l (1) qq (2) Solution: q Aei00 t Be i t
Substitute (2) in (1) 0 g 0 l PCD_STiCM 99 (1) Newtonian (2) Lagrangian g Note! We have not used l ‘force’, ‘tension in the string’ etc. in the g Lagrangian and 0 l Hamiltonian approach!
PCD_STiCM 100 Hamilton’s equations: simple pendulum
L L(,,,) r r T V
1 22 L m mg mg cos 2 L p ml 2 L p 0 r r L H qi - L q i p i - L q i PCD_STiCM 101 L L 2 p ml H qi - L qi L p 0 r r qii p - L 1 22 L m mg mg cos 2 1 H p rp m22 mg mg cos r 2 H mgl( sin ) mgl H p mgl PCD_STiCM 102 simple pendulum L p ml 2 H L p mgl p 0 r r
2 ml p mgl (1) qq g i00 t i t (2) Solution: q Ae Be Substitute (2) in (1) l 0 g 0 PCD_STiCM l 103 g (1) Newtonian (2) Lagrangian l (3) Hamiltonian g 0 l
We have NOT used ‘force’, causality , linear-response 104 PCD_STiCM Lagrangian and Hamiltonian Mechanics has very many applications.
All problems in ‘classical mechanics’ can be addressed using these techniques.
PCD_STiCM 105 However, they do depend on the premise that mechanical system is characterized by position and velocity/momentum, simultaneously and accurately.
PCD_STiCM 106 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 107 (q,p) : How do we get these?
Heisenberg’s
principle of uncertainty
New approach required !
PCD_STiCM 108 ‘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 109 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 110 Mechanical State: | State vectors in Hilbert Space Characterize? Labels?
“Good” quantum numbers/labels | Measurment: C.S.C.O. Complete Set of Commuting Operators Complete Set of Compatible Observables iH| | Evolution of the t Mechanical State of the system Schrödinger EquationPCD_STiCM 111 (,)qq F ma
Galileo Galilei Linear Response. 1564 - 1642 Principle of causality.
Principle of Galileo Newton Variation L(,) q q H(,) q p L d L 0 q dt q HH Lagrange qp, Hamilton PCD_STiCM 112 pqk