Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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Example 8 : Show that the time taken by a particle moving along a curve y = y ( x )
with velocity Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevom the point (0,0) to the point (1,1) is minimum if the curve is a circle having its center on the x-axis. 

Solution: Let a particle be moving along a curve y = y(x) from the point (0, 0) to the point  (1, 1) with velocity  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Therefore the total time required for the particle to move from the point (0, 0) to the point (1, 1) is given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (1) 
where the infinitesimal distance between two points on the path is given by

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Hence the equation (1) becomes 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 

Time t is minimum if the integrand  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 
must satisfy the Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Solving for y′ we get  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Integrating we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Put Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5)

Therefore, 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6)

Squaring and adding equations (5) and (6) we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

which is the circle having center on y –axis. 

 

Example 9 : Show that the geodesic on a right circular cylinder is a helix. Solution: We know the right circular cylinder is characterized by the equations 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (1) 

The parametric equations of the right circular cylinder are 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

where a is a constant. The element of the distance (metric) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

on the surface of the cylinder becomes 

 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

 Hence the total length of the curve on the surface of the cylinder is given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 
 For s to be extremum, the integrand  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

must satisfy the Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (4) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Integrating the equation and solving for z′ we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev(constant). 
Integrating we get  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5)

where a, b are constants. Equation (5) gives the required equation of helix. Thus the geodesic on the surface of a cylinder is a helix. 

 

Example 10 : Find the differential equation of the geodesic on the surface of an inverted cone with semi-vertical angle θ .

Solution: The surface of the cone is characterized by the equation

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (1) 

The parametric equations of the cone are given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (2) 

where for a = sinθ , b = cosθ are constant. Thus the metric 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

on the surface of the cone becomes 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Hence the total length of the curve Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevon the surface of the cone is given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

The length s is stationary if the integrand  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5) 

must satisfy the Euler-Lagrange’s equation

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (7) 

where  c1 = constant. This is the required differential equation of geodesic, and the geodesic on the surface of the cone is obtained by integrating equation (7). This gives

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Example 11 : Find the curve for which the functional  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

can have extrema, given that y(0)=0, while the right –hand end point can vary along 
the line Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Solution:  To find the extremal curve of the functional  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (1) 
we must solve Euler’s equation 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 
where   

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (4) 

This is the second order differential equation, whose solution is given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5)

The boundary condition y ( 0 ) = 0 gives a = 0. 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6) 
The second boundary point moves along the line  Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

where from equation (6) we have y′ = b cos x . Thus Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev gives  

b= 0. This implies the extremal is attained on the line y = 0.

 

Example 12 : If f satisfies Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Then show that f is the total derivative Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevof some function of x and y and conversely. 


Solution:  Given that f satisfies Euler-Lagrange’s equation  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (1) 

We claim that Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

where   Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

As   Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

we write equation (1) explicitly as 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 
We see from equation (2) that the first three terms on the l. h. s. of (2) contain at the highest the first derivative of y. Therefore equation (2) is satisfied identically if the coefficient of y′′ vanishes identically. 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Integrating w. r. t. y′ we get  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Integrating once again we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . (3)

where p ( x, y ) and q ( x, y ) are constants of integration and may be function of x and y only. Then the function f so determined must satisfy the Euler –Lagrange’s equation (1). From equation (3) we find 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev
Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

and

Therefore equation (1) becomes  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev.(4) 

This is the condition that the equation pdx + qdy is an exact differential  
equation dg . 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Therefore, 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5) 
This proves the necessary part.

Conversely, assume that  Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevWe prove that f satisfies the Euler-Lagrange’s equation  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Since    Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Therefore, we find 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Consider now  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

satisfies Euler-Lagrange’s equation. 


Example 13 : Show that the Euler-Lagrange’s equation of the functional 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

has the first integral Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  if the integrand does not depend on x. 
 

Solution: The Euler-Lagrange’s equation of the functional  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

to be extremum is given by 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (1) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

If f does not involve x explicitly, thenExamples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Therefore, we have 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 

Multiply equation (2) by y′ we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (3) 
But we know that 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

From equations (3) and (4) we see that 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (5) 
This is the first integral of Euler-Lagrange’s equation, when the functional 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

 

Example 14 : Find the minimum of the functional

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

if the values at the end points are not given. 

Solution: For the minimum of the functional 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (1) 
the integrand 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (2) 
must satisfy the Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 
Integrating we get  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (5) 
Further integrating we get 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6)

where c, c2 are constants of integration and are to be determined.

However, note that the values of y at the end points are not prescribed. In this case the constants are determined from the conditions. 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (7) 
These two conditions will determine the values of the constants.  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (8) 
where from equation (5) and (6) we have 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

similarly, Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Thus the equations (8) become 

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Solving these equations for cand c2 we obtain Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev

Hence the required curve for which the functional given in (1) becomes minimum is  

Examples : Euler-Lagrange Equation (Part - 2) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (9) 

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