Lagrangian Dynamics Problem solving

7/27/2019 Lagrangian Dynamics Problem solving

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HT2010

First Year Physics: Prelims CP1

Classical Mechanics: Prof. G. Yassin

Problems V Lagrangian Dynamics

The problems are divided into two sections: (A) Standard and (B) Harder.

Section A - Standard Problems

Variational Principles

1. Fermats principle

A light beam is propagating in the x y plane in a media whose refraction

indexndepends only on y.(a) Use Fermats principle to show that the trajectory of the beam from(x0, y0) to (x1, y1) may be obtained by minimizing the functional

S(y) = c1 x1x0

n(y)[(1 + y2)]1

2 dx

wherey =dy/dx and c is the speed of light in vacuum.(b) A light ray propagates from (x0, y0) to (x1, y1) by reflection from

the surface of a flat mirror located at in the plane y = 0 as shown in Fig 1.Show that the angle of reflection r is equal to the angle of incidence i

2. The Brachistochrone Problem-Bernoullis solution

The Brachistochrone Problem was launched by Jean Bernoulli in 1696 andmay be stated as follows: A particle of mass m is descending under theinfluence of constant gravity in a vertical plane. Which path should theparticle follow in order too move from point A to point B in the shortestpossible time?

The solution suggested by Bernoulli recognises that the particle willfollow the path of a beam of light propagating in a dielectric media ofrefractive index n(y), inversely proportional to the particle velocity. .

(a) Given that the optical path will be minimum if Snells law is satisfiedand that the index of refraction is inversely proportional to the free fallvelocityv =

2gy show that the path equation is given by

sin 2gy

= 12ga

,

where is the angle between the ray direction (tangent to the curve) andthe vertical direction.

(b) Eliminate to show that the above equation can be written as:

dy

dx =

a y

y

12

(c) Use the substitution y = a sin2

2 to find that x() =

a

2( sin ),

hence show that the equation of the path is a cycloid.

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Euler-Lagrange Equation

3. Atwoods machine

The three masses shown below move in a vertical plane under the influenceof constant gravity and the tension in the inextensible strings. Assumingthat the pulleys are massless and that all friction forces can be neglected,

(a) write down the constraints equation that result from the fixed length

of the strings, hence show that the motion of the three masses may bedescribed by two generalized coordinates.

(b) Use the E-L equation to find the acceleration of each mass.

(c) Repeat (b) using Newton second law and compare the two methods.Why does the upper pulley rotate despite the fact that the masses on eitherside are equal?

4. Motion in two dimensions

Consider a particle of mass m moving in the x, y plane under the influenceof the potential V(r) where r is the postion vector of the particle in aninertial reference frame.

Construct the Lagrangian and the Hamiltonian of the particle in polar polarcoordinatesr, , hence find which quantities are constants of motion. Is thisconsistent with what you expected from Newtonian mechanics?

5. The simple pendulum

Use E-L equation to calculate the period of oscillation of a simple pendulumof lengthl and bob mass m in the small angle approximation. Assume nowthat the pendulum support is accelerated in the vertical direction at a ratea, find the period of oscillation. For what value ofa the pendulum does notoscillate? Comment on this result.

6. A sliding blockA block of mass m slides on a frictionless inclined plane of mass M, whichitself rests on a horizontal frictionless surface.

(a) Choose the displacement of the inclined plane xand the displacementof the block m s relative to the inclined plane as generalized coordinate andfind the Lagrangian of the system.

(b) Write down the E-L equation for each coordinate and find the ac-celeration of the inclined plane. Compare this solution with the one youobtained using Newton laws.

7. Rotating bead

A bead of mass m is constrained to slide on a frictionless wire which ismade to rotate about a vertical axis at an angular velocity . The wire istilted away from the vertical by an angle and the location of the bead ismeasured by the coordinater .

(a) Write down the equation of motion of the bead using the E-L equa-tion. Test the integrity of your equation by taking extreme values of.

(b)Find the general solution assuming that at t = 0, r = r0,r = 0.Based on this solution, show that for r0 = gcos/

2sin, the bead movesin circular motion (as expected!). Describe the motion for r < r0and r > r0.

(c) Which of the following quantities is a constant of the bead motion:angular momentum with respect to the origin, the Hamiltonian, total en-ergy?

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8. Sliding on a sphere

A particle of massmslides without friction down the surface of a hemisphereof radius R.

(a) Construct the Lagrangian of the problem in terms of the polar co-ordinates(r, ), in the range when the constraint r = R is valid. Find theequation of motion.

(b) Allow the radius of the sphere to vary by an infinitesimal amountand write the equation of morion with r as a free variable. Include thepotentialV(r)of the rection force applied by the hemisphere on the object.Write the new Lagrangian and find the reaction force for r = R. Comparewith the result derived from NII.

(c) Assuming that the particle is released from the top of the sphere fromrest, show that the particle leaves the surface at an angle cos max= 2/3.

9. A beed on a rotating hoop

A vertical circular hoop of radius R rotates about a vertical axis at anangular velocity. A bead of massmcan slide on the hoop without frictionand is constrained to stay on the hoop. By taking the angle between the

radius line and the vertical, as a generalized coordinate,(a) Find the Lagrangian and the equation of motion. Using the conceptof effective potential or otherwise, find the three equilibrium positions ofthe bead.

(b) Discuss the stabiliy of each equiibrium point and find the frequencyof smal oscillations about the stable ones.

(b) Find the Hamiltonian and the total energy T+V. Is either of thema constant of motion?

Section B - Harder Problems

10. 2-D spring

A particle of mass m is attached to the free end of a massless spring ofequilibrium length a and spring constant k. The other end of the springis pivoted to a frictionless horizontal surface and the particle is allowed tomove in 2-D under the influence of the spring force which is assumed toobey Hooks law.

(a) Write the E-L equations for the polar coordinate (r, ). Identify thecyclic coordinates and the corresponding conserved quantities. Write downthe equation of motion in terms of the variabler.

(b) Write the total energy of the system (for a given angular momentumJ ) and analyse the motion using the concept of effective potential. Findthe radius for circular orbit and show that it is consistent with the valueobtained from Newton laws.

(c) Use Newton II in Cartesian coordinates to show that if the rest lengthof the spring is negligible (a 0), the path of the particle is elliptical withrmeasured from the centre of the ellipse. Use the expression for total energyEto find the major and the minor axes of the ellipse.[Answer: a, b= r0/(

1 )where r0 = J2/mEand =

1 kJ2/mE2)]

11. The Brachistochrone Problem

A particle of mass m is constrained to slide without friction, in a verticalplane, down a cycloidal surface which may be parameterized as:

x= a( sin), y= a(1 + cos ).

Att = 0 the partical was at rest at the origin A.(a) Show that the equation of the surface describes the motion of a

particle on the rim of a rolling wheel.

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(b) Write down the Euler-Lagrange equation for the coordinates . Bymaking the substitutionu = cos

2, show that the coordinate of the particle

u executes simple harmonic oscillations with a period independent of theamplitude.

(c) Compare the time that takes the particle to move from A to thebottom of the cycloid B with the time that takes it to move from Ato B along a straight line. Discuss the link with the Brachistochrone

problem. [Answers: (b) T = 2(4a/g)1

2 , (d) tline = (1 + 4/2

)1

2 )tcyc]

12. A Pendulum with accelerated support

A box of mass Mcan slide horizontally on a frictionless surface. A simplependulum of string length l and mass m, is suspended inside the block.Denote the coordinate of the centre of mass of the box by x and the anglethat the pendulum makes with the vertical by . At t = 0 the pendulumdisplacement is = 0= 0

(a) Find the Lagrangian and the equation of motion for the generalizedcoordinates x and . Which conservation law is obtained as a result of thecyclic coordinate?

(b) Find the solutions for x and in the small angle approximation,

hence show that the pendulum and the box execute SHO about their centreof mass at a frequency

=

M+ m

M

12g

l

12

13. Normal modes

Two equal masses m1 =m2 are connected by two massless springs of forceconstantsk1 and k2 as shown below. The system is placed on a horizontalfrictionless table and attached to the wall.

(a) Write the Lagrangian of the system using the coordinates x1 and

x2 that give the displacements of the masses m1 and m2 respectively fromtheir equilibrium positions. Use the E-L equations to find the equation ofmotion of each mass.

(b) Find the solutions in which the two masses execute simple harmonicoscillations at the same frequency (normal modes). Verify your solutions byconsidering the special cases k1 k2 andk1 k2.

l

r

14. The Spherical Pendulum

Consider a spherical pendulum which consists of a mass m suspended by amessless string of length l from the ceiling as show below.

(a) Write the Lagrangian of the system in terms of the polar coordinats

(, ). Write the Hamiltonian and show thatH=T+ V.(b) Show that is a cyclic coordinate and find the corresponding con-served quantity. Hence show that the Hamiltonian may be written as

H= 1

2ml22 + Veff()

and find an expression for Veff().(c)SketchVeff()and show that the pendulum can move steadily around

a circle with = 0 at an angular velocity given by:

2 = g

l cos 0

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16. Kepler laws for two-body system.

Consider an isolated system of two stars of massesm1andm2 and positionvectors r1 and r2, moving under their mutual gravitation. Show that in thecentre of mass frame, the Lagrangian of the system is equal to that of asingle particle of mass = m1m2

m1+m2and position vector r= r1 r2. Deduce

therefore that the procedure employed in Kepler problem may be used topredict the motion of each star. Using the solution for a an object of massmorbiting a much heavier star of mass M m, derive expressions for theradial equation, the effective potential and Kepler third law.

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Lagrangian Dynamics Problem solving

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