The principle of least action in classical mechanics

1. The principle of least action
The principle of least action
Ipsita Mandal

2. Throwing A Stone Upwards
F = m a  y = g t1 ( t2 – t ) / 2
Parabola
t1
t2
t
y
Previous
classes
Newton’s 2nd
law gives the right answer

3. Alternate Approach: Principle Of Least Action
t1
t2
t
y
Correct
path
Incorrect
path

4. Alternate Approach: Principle Of Least Action
t1
t2
t
y
Correct
path
Incorrect
path
More fundamental
than Newton’s laws

5. Throwing A Stone Upwards
t1
t2
t
y

6. Throwing A Stone Upwards
Possible paths
with same
y(t1) & y(t2)
t1
t2
t
y

7. Quick Check
t1
t2
t
y
Is ´(t) an allowed path?
´(t)

8. t1
t2
t
y
´(t) not an
allowed path
´(t)
Quick Check
Path should be single-valued function of t
Arrow of time should be respected
´(t) is multi-valued
e.g. at times ta & tb
 in some regions
particle moves
backwards in time!
ta
tb

9. Alternate Approach: Principle Of Least Action
Total [ Kinetic Energy (.T.) - Potential Energy (.U.) ]
over the path
is as small as possible for the actual path of an object
going from one point to another

10. Alternate Approach: Principle Of Least Action
Total [ Kinetic Energy (.T.) - Potential Energy (.U.) ]
over the path
S =
True path is the one for which S is least
Action

11. Throwing A Stone Upwards
T =
U =
S =
True path is the one for which S is least
S2 > S1
S1
S2

12. Extremization Problem
Calculus of Maxima & Minima?
Applies when we have a function f of some variables
and we have to find the values of those variables
where f is most or least
E.g. Find the minimum value of the curve f(x)
f(x)
x x0
minimum

13. Extremize Funtionals
Calculus of Maxima & Minima?
Applies when we have a function f of some variables
and we have to find the values of those variables
where f is least or most
Here, for each path we have a number ( S ) and we
have to find the path for which S is the minimum
Need to minimize a functional
function of function(s)
S[y(t)]

14. Minimize Action S
Brute Force:
Calculate S for millions and millions of paths and
look at which one is lowest

15. Minimize Action S
Brute Force:
Calculate S for millions and millions of paths and
look at which one is lowest
Calculus of Variations:
For the true path y0(t), a curve which differs only a
little bit from it will have first order variation in S
zero

16. Minimize Action S
y0(t)
y0(t) + ² ´(t)
t1
t2
t
y
For the true path, a neighbouring path will have
no first order variation in S
Proof:
S[y0(t)] = Action of true path y0(t)
S[y0(t)+ ² ´(t)] = Action of trial path y0(t) + ² ´(t)
➢² is an infinitesimally small number
➢´(t) is an arbitrary single-valued
non-singular function of t
satisfying ´(t1) = ´(t2) = 0

17. Minimize Action S
y0(t)
y0(t) + ² ´(t)
t1
t2
t
y
For the true path, a neighbouring path will have
no first order variation in S
Proof:
S[y0(t)] = Action of true path y0(t)
S[y0(t)+ ² ´(t)] = Action of trial path y0(t) + ² ´(t)
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)] ∝ ²
to first order in ²

18. Minimize Action S
y0(t)
y0(t) + ² ´(t)
t1
t2
t
y
For the true path, a neighbouring path will have
no first order variation in S
Proof:
S[y0(t)] = Action of true path y0(t)
S[y0(t)+ ² ´(t)] = Action of trial path y0(t) + ² ´(t)
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)] ∝ ²
to first order in ²
±S flips sign if ² flips sign
S[y0(t)] is least  ±S = 0 + O(²2)

19. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

20. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

21. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

22. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

23. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

24. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]
Rearrange the term with d´/dt
to make it have an ´ using
Integration by parts

25. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]
Rearrange the term with d´/dt
to make it have an ´ using
Integration by parts

26. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]

27. y0(t)
y0(t) + ² ´(t)
t1
t2
Compute Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]
total
derivative

28. Zero Variation
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]
y0(t)
y0(t) + ² ´(t)
t1
t2
when S[y0(t)] is least

29. y0(t)
y0(t) + ² ´(t)
t1
t2
Equivalence To Newton’s 2nd
Law
±S ≡ S[y0(t)+ ² ´(t)] – S[y0(t)]
Newton’s 2nd
law
when S[y0(t)] is least

30. Generalizations
➢ Extend to 2 or 3 dimensions  e.g. S[x(t), y(t), z(t)]

31. Generalizations
➢ Extend to 2 or 3 dimensions  e.g. S[x(t), y(t), z(t)]
➢ Extend to many particles  e.g. S[y1(t), y2(t)]

32. Generalizations
➢ Extend to 2 or 3 dimensions  e.g. S[x(t), y(t), z(t)]
➢ Extend to many particles  e.g. S[y1(t), y2(t)]
➢ Least action formalism analogous to Fermat’s principle of
least time for light propagation (homework)

33. Generalizations
➢ Extend to 2 or 3 dimensions  e.g. S[x(t), y(t), z(t)]
➢ Extend to many particles  e.g. S[y1(t), y2(t)]
➢ Least action formalism analogous to Fermat’s principle of
least time for light propagation (homework)
➢ Action principle applies to other branches like relativity &
quantum mechanics  it is a fundamental principle

34. Generalizations
➢ Extend to 2 or 3 dimensions  e.g. S[x(t), y(t), z(t)]
➢ Extend to many particles  e.g. S[y1(t), y2(t)]
➢ Least action principle analogous to Fermat’s principle of
least time for light propagation (homework)
➢ Action formalism applies to other branches like relativity &
quantum mechanics  it is a fundamental principle
➢ Generically the function that is integrated over time to get
the action is called the Lagrangian

35. Is It A Minimum?
±S = 0 implies S[correct path] is extremum
“Least Action” is a misnomer
Correct: S[correct path] is never a maximum

36. Is It A Minimum?
➢ Red path stays very close to correct path,
but oscillates
➢ Since y is almost the same, U(y) is almost
the same for both
➢ Red path has higher T = ½ m (dy/dt)2
➢ We can always construct a path for which
S is larger
y0(t)
y0(t) + ² ´(t)
t1
t2
±S = 0 implies S[correct path] is extremum
“Least Action” is a misnomer
Correct: S[correct path] is never a maximum

37. ➢ The Feynman Lectures On Physics Vol II, Chapter 19
➢ Lecture by K. Young at the Physics Department of The
Chinese University of Hong Kong
(https://www.youtube.com/watch?v=IhlSqwZBW1M)
➢ Mechanics: Volume 1 by L. D. Landau & E. M. Lifshitz,
Chapter 1
References