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Lec 3 SEGP

The document discusses the Hamiltonian function H and its relation to the Lagrangian L. It defines H as the sum of generalized momenta pk multiplied by generalized velocities qk minus the Lagrangian L. Hamilton's equations relate the time derivatives of generalized coordinates and momenta to partial derivatives of H with respect to the coordinates and momenta. For conservative systems where potential energy V does not depend on generalized velocities, the Hamiltonian H represents the total energy of the system, which is the sum of kinetic energy T and potential energy V. The document also provides examples of applying the Euler-Lagrange equation to find equations for the shortest distance between two points and the curve that gives a surface of revolution with minimum area.

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Rahul
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679 views11 pages

Lec 3 SEGP

The document discusses the Hamiltonian function H and its relation to the Lagrangian L. It defines H as the sum of generalized momenta pk multiplied by generalized velocities qk minus the Lagrangian L. Hamilton's equations relate the time derivatives of generalized coordinates and momenta to partial derivatives of H with respect to the coordinates and momenta. For conservative systems where potential energy V does not depend on generalized velocities, the Hamiltonian H represents the total energy of the system, which is the sum of kinetic energy T and potential energy V. The document also provides examples of applying the Euler-Lagrange equation to find equations for the shortest distance between two points and the curve that gives a surface of revolution with minimum area.

Uploaded by

Rahul
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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The Hamiltonian function H,

The Lagrangian, in general is defined as ,

L L qk , q k , t
Let us introduce a new function H, known as
Hamiltonian, a function (p, q, t) and defined as

H pk q k L qk , q k , t
k

Hamiltons equations or Hamiltons canonical equation


of motion
H H
q k , p k ,
pk qk
L H
and also ,
t t
Physical significance of the Hamiltonian function :-

(i). If L is not an explicit function of time , then Hamiltonian H


is constant of motion i.e.
L H
0
t t
or H const.

(ii). For conservative systems, the potential energy does not


depend upon generalized velocity, i.e
V
0
q k
and also we know that
H pk q k L qk , q k , t
k
L
H q k L qk , q k , t
k q
k
(T V )
q k L qk , q k , t
k q k

2

1 mq k2
q k L qk , q k , t
k q k
2T L
T V E (total energy )

Thus for conservative systems, the Hamiltonian function


H represents the total energy of the system.
Variational principles :
(i) Euler-Lagrange differential equation for one dependent
and one independent variable :-
(i) Variational principle give the necessary conditions that
the quantity appearing as an integral has either a
minimum or maximum i.e. stationary value.
Consider the simplest integral
x2
J f ( y, y ' , x)dx
x1

dy
such that y'
dx
The dependence of y on x is not specified initially.
We have to choose the path of integration i.e. y(x) such that
J has stationary value.

Let us consider two paths out of infinite number of possibilities


such that the difference between these two for the given value
of x is the variation of y.
We get the condition that the integral

x2
J f ( y, y ' , x)dx
x1

has an extremum value is given by


d f f
0
dx y ' y
This is known as Euler-Lagrangian Differential equation.
Simple application of Euler-Lagrangian Differential
equation :-

(i). Prove that the shortest distance between two


points in a plane is a straight line :-

An element of distance between two points in a plane is a straight


line is given by
ds dx
1/ 2
2
dy 2

2 1/ 2
dy
1 dx
dx

The total distance between two points having co-ordinates (x1, y1)
and (x2, y2) is given by
2 1/ 2
x 2, y 2 x 2, y 2 dy
J ds 1 dx
x1, y1 x1, y1
dx

By comparing this equation with


x2
J f ( y, y ' , x)dx
x1

dy
such that y'
dx
We get,
2 1/ 2
dy 2 1/ 2
f ( y , y ' , x ) 1
1 y'
dx
If J is to be minimum, it must be satisfied Euler- Lagrangian
Equation. Therefore,
d 2 1/ 2 2 1/ 2
(1 y ' ) (1 y ' ) 0
dx y ' y
d y'
or 2 1/ 2
0
dx (1 y ' )
y'
or 2 1/ 2
c
(1 y ' )
c c
or y' 2 1/ 2
a ( where a 2 1/ 2
is const )
(1 c ) (1 c )
dy
or a
dx
or y ax b
Where, b is constant of integration.
Which is the equation of a straight line. Hence Eulers equation
predicts that the shortest distance between two fixed points in a
plane is a straight line.
(ii). Equation of a curve which when revolved about y axis
gives minimum surface of revolution.

Let us suppose we form surface of revolution by taking some


curves passing between two fixed end points (x1, y1) and (x2,
y2) defining the x-y plane , and revolving it about the y-axis
as shown in figure. The area2
of a strip of the surface is
2xds 2x 1 y ' dx and the total area
2
is 2 x 1 y '2 dx
1

By comparing this equation with


x2
J f ( y, y ' , x)dx
x1

dy
such that y'
dx
We get,

2 1/ 2

f ( y, y ' , x) x 1 y '
and
If J is to be minimum, it must be satisfied Euler- Lagrangian
Equation. Therefore,
d 2 1/ 2 2 1/ 2
x (1 y ' ) x (1 y ' ) 0
dx y ' y
d xy '
or 2 1/ 2
0
dx (1 y ' )
xy ' 2 2 2 2
or a or y ' ( x a ) a
(1 y '2 )1/ 2
dy a
or y ' 2
dx ( x a 2 )1/ 2
The general solution of this differential equation is

dx x
y a 2 2 1/ 2
b a arccos h b
(x a ) a
y b
or x a cosh
a

Which is the equation of catenary.

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