This document presents Hamilton's variational principle for conservative systems and derives the Lagrangian equations of motion from it. It states that Hamilton's principle is that the motion of a system from time t1 to t2 is such that the line integral of the Lagrangian L = T - V is an extremum. It then proves this principle using the Alembert principle and shows that requiring the variation of the line integral to be zero under independent variations of the generalized coordinates yields the Lagrange equations of motion. Finally, it concludes that the Lagrange equations directly follow from Hamilton's variational principle.

1. PHYSICS DEPARTMENT
PH-201 CLASSICAL MECHANICS
HAMILTON’S VARIATIONAL PRINCIPLE
Submitted to: Submitted by:
Dr. Uttam Paliwal Praveen Dadhich
M.Sc. Physics(2nd sem.)
JAI NARAIN VYAS
UNIVERSITY
JODHPUR

2. STATEMENT:- Hamilton’s variational principle for conservative
system stated as follows:
The motion of system from time t1 to time t2 is such that the
line integral
Where L = T-V, is an extremum for path of motion. i.e.
HEMILTON’S VARIATIONAL PRINCIPLE

3. Proof :- From Alembert principle
We know that 𝜹 can be interpreted as an operator which produces a
small change in ri not connected with time. It can be treated like a
different operator both 𝜹 ,d operators can commute , so we can write

5. Integrating the above eq. with respect to the time from initial instant t1 to final
instant t2 of the motion, we get
But at t1 and t2
So above equation will be
Which is Hamilton’s variation principle.

6. LAGRANGIAN EQUATION OF MOTION
FROM HAMILTON’S PRINCIPLE :-
According to Hamilton's variation principle motion of a
conservative system from time t1 to time t2 is such that the
variation of the line integral
dt
t
q
q
L
I
t
t
j
j



2
2
)
,
,
(
 
t
q
q j
j ,

  
t
q
q j
j ,




  dt
t
t
L
q
q
L
q
q
L
I j
j
j
t
t j j
]
)
(
[
2
1
























 

7. = dt
q
q
L
dt
d
q
q
L
dt
q
q
L j
j
t
t
t
t
j
j
j
t
t j j 

 














 
  )
(
]
[
]
[
2
1
2
1
2
1
= ]
)}
(
{(
[
2
1
dt
q
q
L
dt
d
q
L j
j
t
t j j 








 






I
I 0

I

,

8. dt
q
q
L
dt
d
q
L
j
j
t
t j

}
(
{
2
1







=
Since qj are independent of each other, the variations will be
independent .Hence If and only if the coefficients
separately vanish,
j=1,2,3……..n
Which are Lagrange’s equation of motions for a conservative
system. It is thus obvious that these equations follow directly from
Hamilton’s variational principle.
0
)
( 






j
j q
L
dt
d
q
L
0

I


9. REFERENCE:-CLASSICAL MECHANICS
(GUPTA KUMAR SHRMA)
THANK YOU……….