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Lagrange, 1736-1831

What Lagrange did?

---- Formulated and interpreted D Alemberts principle ingeneralized coordinate.

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APPLICATIONS OF LAGRANGEs EQUATION

---- (1) --- (2)

For any given physical system

(Not necessarily a mechanical system):

1. Step 1: Locate generalized coordinate/s.2. Step 2: Express K. E. and P. E in terms of g.c.

3. Step 3: Determine L = T-V

4. Step 4: Use equation (1) if there is a force or (2) if force free.

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Example 4.1: (By Lagrangess equation)

Q. Determine equation of motion of an idea Atwoods machine

Answer (hints)

Let g. C. Be x (position of m1).

Follow the steps (1-3).

Use equation (2)

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Example 4.2: (By Lagrangess equation)

Q. A] Analyze variation of charge with time in LC circuit. (Assumption:

Charge does not flow at time t = 0).

Answer (Hints)

Take q as g.c.

Write expression of K. E. And P. E. In

Terms of L and C.

Use equation 2.

B. Show that S. H. O. motion is a mechanical analog of the above.

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If there is a dissipation in the system.

Examples:

(i) Ball falls under gravity in a liquid.

(ii) A resistance in LR circuit.

(iii) Damped harmonic oscillator.

MechanicalAnalog

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Example 4.3: (Particle in an Electromagnetic field)

Q. Generalized potential or velocity dependent potentialis applicableTo a very important force field, namely electromagnetic forces of moving

charges. From x component of the Lorentz forceF = q [E + (v x B)

show that, the potential energy is

U = q - qA.v [Where, = (x,y,z,t) and A = A (x,y,z,t) are scalar and vectorpotential respectively]

What is the Lagrangian of the system?

Example 4.4: (Generalized kinetic energy)

Q. A] Derive an expression for generalized kinetic energy. Show that if transformation

Equations do not contain time explicitly, then K. E. Is always a homogeneous,

quadratic form of generalized velocities.

B] Show an example in case of time dependent transformation.

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Example 5 (On degrees of freedom)

Q. A] To describe a rigid body in space, one must know three non collinearpoints on it. The constraint condition is that the distance between two

points is constant. Calculate number of degrees of freedom.

B] If now one point is made fixed in space, what is the degree of freedom?

Ans. (Hints)

Write down equations of distances for three points and use constraint

Conditions [Constraint equations].

No. of d.o.f. = 9 -3 = 6

The motion of a rigid body can always be understood as a translation of any of its points relative

to an inertial system and a rotation of the body about this point. -- M. Chasles

Now, if one point is fixed, d. o. f. = 6-3=3