The document discusses the principle of least action, which states that among all possible paths that an object can take between two points, the actual path taken is the one that minimizes the action. The action is defined as the time integral of the Lagrangian, which is the kinetic energy minus the potential energy. This principle can be derived using the calculus of variations and results in equations equivalent to Newton's second law. The principle of least action is more fundamental and can be generalized to multiple dimensions, particles, and other areas of physics such as relativity and quantum 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