The document discusses solutions to the Hamilton-Jacobi equation, emphasizing Hamilton's principal function and characteristic function as two key types of solutions. It describes the equation's formulation as a first-order partial equation involving multiple independent variables and transformation equations that yield constant quantities. Additionally, it highlights the physical significance of these solutions in relation to the Lagrangian and energy conservation within conservative systems.

1. SESSION – 2022 - 23

2. HAMILTON’S PRINCIPLE AND
CHARACTERISTIC FUNCTION

3. SOLUTION OF HAMILTON-JACOBI EQUATION
The Hamilton Jacobi equation is given by
0
)
,
,
(
,
,...
,
,
,...
,
2
2
2
2
1
2
2
1 

















t
t
P
q
F
t
q
F
q
F
q
F
q
q
q
H
j
j
n
n
Solution of Hamilton’s-Jacobi equation is to obtain transformation equation in which co-
ordinate of new set be constant quantities.
Since these co-ordinates , being constant quantities , can be taken as initial values .
Hence the solutions are –
)
,
,
(
)
,
,
(
q 0
0
t
q
q
t
p
q
q
j
j 




4. We have two solutions for this equation-
1. Hamilton’s Principal Function
2. Hamilton’s Characteristic Function
HAMILTON’S PRICIPLE FUNCTION :--
The H-J equation has the form of first order partial equation in (n+1) independent variables.
Consequently complete solution must involve (n+1) independent constants of integration ........
,
, 3
2
1 


Hence a complete set of solution of H-J equation is –
)
,
,...,
,
,...,
(
S
)
,
,
(
)
,
,
(
F
1
1 t
P
P
q
q
S
t
P
q
S
S
t
q
S
n
n
j
j
j
j


 
These n constants can be taken as new momenta j
P

5. S is similar to the function therefore n transformation equations can be written as -
)
,
,
(
S t
q
S j
j 

)
,
,
(
F2 t
P
q j
j
)
,
,
(
)
,
,
(
2
2
j
j
j
j
j
j
j
j
j
j
P
t
P
q
S
P
F
Q
q
t
P
q
S
q
F
p












t
S
q
p
t
S
S
q
p
t
S
S
q
q
S
j
j
j
j
j
j
n
i j
j
n
i j
































0
dt
dS
eq.1
from
1
1



……..(1)
……..(2)
……….(3)
........(4)

6. S is the solution of H-J equation therefore it will satisfy it
0
,
,...
,
,
,...
,
2
1
2
1 

















t
S
t
q
S
q
S
q
S
q
q
q
H
n
n
H
t
S
t
S
H 







 0
Substituting in (4)
constt
Ldt
S
)
,
,
(
dt
dS







t
q
p
H
q
p j
j
Physical significance :- time integral of Lagrangian is equal to S.

7. H-J equation for conservative system and Hamilton's characteristic function
For conservative system H does not include time
 
)
(
energy
total
represents
particular
a
here
)
variables
of
n
(separatio
)
,
(
)
,
,
(
where
0
)
,
,
(
,...,
,
,...,
becomes
equation
J
-
H
,
,
1
1
1
1
1
1
H
t
S
q
W
q
S
t
q
W
t
q
S
P
t
t
q
S
q
S
q
S
q
q
H
q
S
q
H
p
q
H
H
j
j
j
j
j
j
j
j
j
j
n
n
j
j
j
j

















































Where, W = Hamilton’s characteristic
function

8.  
























































constant
action
)
0
(
)
,
(
,
equation
J
-
H
system
ve
conservati
for
so
0
,
-
:
get
we
values
the
ng
substituti
on
1
1
constt
dt
q
p
W
q
p
q
q
W
W
q
q
W
dt
dW
q
W
W
q
S
q
W
q
W
q
H
q
W
q
H
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j







……..(A)
Physical Significance :- The time integral of Lagrangian is equal to the ‘W’ .

9. REFERENCE BY :
1. CLASS NOTES
2. BOOK - GUPTA KUMAR

10. THANK
YOU...!