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

Physics for IIT JAM, UGC - NET, CSIR NET

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

The document Examples : Euler-Lagrange Equation (Part - 1) - 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 1 : Show that the geodesic (shortest distance between two points) in a Euclidian plane is a straight line.

Solution: Take P ( x1 , y1 ) and Q ( x2 , y2 ) be two fixed points in a Euclidean plane. Let y = f ( x ) be the curve between P and Q. Then the element of distance between two neighboring points on the curve y = f ( x ) joining P and Q is given by

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev
Hence the total distance between the point P and Q along the curve is given by 

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

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

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Now from equation (2) we find that  

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

Integrating we get  

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

Squaring we get  

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

Integrating we get  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4)

This is the required straight line. Thus the shortest distance between two points in a Euclidean plane is a straight line. 


Example 2 : Show that the shortest distance between two polar points in a plane is a straight line. 

Solution:  Define a curve in a plane. If A ( x, y ) and B ( x + dx, y + dy ) are infinitesimal points on the curve, then an element of distance between A and B is 
given by 

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

Let  θ = θ ( r ) be the polar equation of the curve and Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev be two polar points on it.  Recall the relations 

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

Hence equation (1) becomes 

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

Thus the total distance between the points P and Q becomes 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

The functional I is shortest if the integrand 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (4) 

must satisfy the Euler-Lagrange’s differential equation 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev        ...(5) 

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

Squaring and solving for θ we get 

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

On integrating we get 

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

where θ0 is a constant of integration. We write this as  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6) 

This is the polar form of the equation of straight line. Hence the shortest distance between two polar points is a straight line. 

Note : If r = r (θ ) is the polar equation of the curve, then the length of the curve is given by 

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

Since the integrandExamples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevdoes not contain θ , we therefore have 

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

Solving this equation we readily obtain the same polar equation of straight line as the geodesic. 

 

Example 3 : Show that the geodesic Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevon the surface of a sphere is an arc of the great circle. 

Solution : Consider a sphere of radius r described by the equations

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

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

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev be two neighboring points on the curve joining the points P and Q. Then the infinitesimal distance between A and B along the curve is given by 

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

where from equation (1) we find 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev
Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (3) 

Squaring and adding these equations we readily obtain   

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

Hence the total distance between the points P and Q along the curve φ = φ (θ ) is given by  

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

where  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6) 

The curve is geodesic if the functional I is stationary. This is true if the function f must satisfy the Euler-Lagrange’s equations.

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (7) 

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

Integrating we get  

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

Solving for φ ′ we get  

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

On simplifying we get  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (8) 

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

Therefore we have  

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

Integrating we get 

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

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

This is the first-degree equation in x, y, z, which represents a plane. This plane passes through the origin, hence cutting the sphere in a great circle. Hence the geodesic on the surface of a sphere is an arc of a great circle. 

 

Example 4 : Show that the curve is a catenary for which the area of surface of revolution is minimum when revolved about y-axis.

Solution: Consider a curve between two points ( x1 , y1 ) and ( x2 , y2 ) in the xy plane whose equation is y = y ( x ) . We form a surface by revolving the curve about y-axis. Our claim is to find the nature of the curve for which the surface area is minimum. Consider a small strip at a point A formed by revolving the arc length ds about y –axis. If the distance of the point A on the curve from y-axis is x, then the surface 

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

area of the strip is equal to 2π x ds . But we know the element of arc ds is given by 

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

Thus the surface area of the strip ds is equal to 

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

Hence the total area of the surface of revolution of the curve y = y ( x ) about y- axis is given by 

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

This surface area will be minimum if the integrand 

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

must satisfy Euler-Lagrange’s equation

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

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

Integrating we get 

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

Solving for y′ we get  

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

Integrating we get 

 Or  Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (4) 

This shows that the curve is the catenary

 

  • The Brachistochrone Problem :  

The Brachistochrone is the curve joining two points not lie on the vertical line, such that the particle falling from rest under the influence of gravity from higher point to the lower point in minimum time. The curve is called the cycloid. 

Example 5: Find the curve of quickest decent. 

 Or   

  A particle slides down a curve in the vertical plane under gravity. Find the curve such that it reaches the lowest point in shortest time. 

Solution: Let A and B be two points on the curve not lie on the vertical line. Let Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRevbe the speed of the particle along the curve. Then the time required to fall an 
arc length ds is given by 

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

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

Therefore the total time required for the particle to go from A to B is given by

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

Since the particle falls freely under gravity, therefore its potential energy goes on decreasing and is given by 

V = −mgx , 
and the kinetic energy is given by 

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

Now from the principle of conservation of energy we have  

T + V = constant.

Initially at point A, we have x = 0 and v = 0 . Hence the constant is zero.   

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

Hence equation (1) becomes 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Thus tAB is minimum if the integrand  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

must satisfy Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (5) 

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

Integrating we get 

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

Solving it for y′ we get  

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

Integrating we get 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (6) 

Put  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (7) 

Hence    

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

hence  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (8) 

Thus from equations (7) and (8) we have  

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

This is a cycloid. Thus the curve is a cycloid for which the time of decent is minimum. 

 

Example 6 : Find the extremal of the functional   

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

subject to the conditions that  

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

Solution:  Let the functional be denoted by 

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

The functional is extremum if the integrand  

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

must satisfy the Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (3) 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

This is second order differential equation, whose complementary function (C.F.) is given by 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev . . . (5)

where c1 and c2 are arbitrary constants. The particular integral (P.I.) is  

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

Hence the general solution is given by 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (6)

This shows that the extremals of the functional are the two-parameter family of curves. On using the boundary conditions we obtain 

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

Hence the required extremal is 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (7) 

 

Example 7 : Find the extremal of the functional  

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

subject to the conditions that  

y (1) = 0, y ( 2 ) = 3 .

Solution:  Let the functional be denoted by 

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

The functional is extremum if the integrand  

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

must satisfy the Euler-Lagrange’s equation 

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev     (3) 

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

Integrating we get  

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

Integrating we get  

Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev. . . (4) 

Now using the boundary conditions we get  

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

Solving these two equations we obtain  

a = 2, b = −1 .  

Hence the required functional becomes 
Examples : Euler-Lagrange Equation (Part - 1) - Classical Mechanics, CSIR-NET Physical Sciences Physics Notes | EduRev  . . . (5) 

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