Chapter7 Lagrangian and Hamiltonian Mechanics

1 Chapter 7 Lagrangian and Hamiltonian MechanicsAbstract Chapter7is devoted to problems solved by Lagrangian and Basic Concepts and FormulaeNewtonian Mechanics deals with force which is a vector quantity and therefore dif-ficult to handle. On the other hand, Lagrangian Mechanics deals with kinetic andpotential energies which are scalar quantities while hamilton s equations involvegeneralized momenta, both are easy to handle. While Lagrangian Mechanics con-tains n differential equations corresponding to n generalized coordinates, Hamil-tonian Mechanics contains 2n equation, that is, double the number. However, theequations for Hamiltonian Mechanics are symbol q is a generalized coordinate used to represent an arbitrary coordi-nate x, , , T is the kinetic energy, V the potential energy then the Lagrangian L isgiven byL= T V( ) Lagrangian Equation:ddt dLd qK L qK= 0(K= 1, )( )where it is assumed that V is not a function of the velocities, v qK= 0.

2 Eqn (2)is applicable to all the conservative n independent coordinates are required to specify the positions of themasses of a system, the system is of n degrees of rs=1ps qs L( )where psis the generalized momentum and qKis the generalized Lagrangian and Hamiltonian MechanicsHamiltonian s Canonical equations H pr= qr, H qr= pr( ) Consider a particle of mass m moving in a plane under the attractive force m/r2directed to the origin of polar coordinates r, . Determine the equationsof (a) Write down the Lagrangian for a simple pendulum constrained to move ina single vertical plane. Find from it the equation of motion and show thatfor small displacements from equilibrium the pendulum performs simpleharmonic motion.

3 (b) Consider a particle of mass m moving in one dimension under a force withthe potential U(x)= k(2x3 5x2+ 4x), where the constant k> 0. Showthat the point x= 1 corresponds to a stable equilibrium position of theparticle. Find the frequency of a small amplitude oscillation of the particleabout this equilibrium position.[University of Manchester 2007] Determine the equations of motion of the masses of Atwood machine by theLagrangian Determine the equations of motion of Double Atwood machine which consistsof one of the pulleys replaced by an Atwood machine. Neglect the masses A particular mechanical system depending on two coordinates u andv haskinetic energy T=v2 u2+ 2 v2, and potential energy V= u2 v2.

4 Writedown the Lagrangian for the system and deduce its equations of motion (do notattempt to solve them).[University of Manchester 2008] Write down the Lagrangian for a simple harmonic oscillator and obtain theexpression for the time A particle of mass m slides on a smooth incline at an angle . The incline is notpermitted to move. Determine the acceleration of the A block of mass m and negligible size slides on a frictionless inclined plane ofmass M at an angle with the horizontal. The plane itself rests on a smoothhorizontal table. Determine the acceleration of the block and the inclined A bead of mass m is free to slide on a smooth straight wire of negligible masswhich is constrained to rotate in a vertical plane with constant angular speed about afixed point.

5 Determine the equation of motion andfind the distance xfrom thefixed point at time t. Assume that at t= 0 the wire is Consider a pendulum consisting of a small mass m attached to one end of aninextensible cord of length l rotating about the other end which isfixed. Thependulum moves on a spherical surface. Hence the name spherical inclination angle in the xy-plane can change independently.(a) Obtain the equations of motion for the spherical pendulum.(b) Discuss the conditions for which the motion of a spherical pendulum isconverted into that of (i) simple pendulum and (ii) conical Two blocks of mass m and M connected by a massless spring of spring con-stant k are placed on a smooth horizontal table.

6 Determine the equations ofmotion using Lagrangian A double pendulum consists of two simple pendulums of lengths l1and l2and masses m1and m2, with the cord of one pendulum attached to the bobof another pendulum whose cord is fixed to a pivot, Determine theequations of motion for small angle oscillations using lagrange s Use hamilton s equations to obtain the equations of motion of a uniformheavy rod of mass M and length 2a turning about one end which A one-dimensional harmonic oscillator has Hamiltonian H=12p2+12 down Hamiltonian s equation andfind the general Determine the equations for planetary motion using hamilton s Two blocks of mass m1and m2coupled by a spring of force constant k areplaced on a smooth horizontal surface, Determine the natural fre-quencies of the Lagrangian and Hamiltonian A simple pendulum of length l and mass m is pivoted to the block of mass Mwhich slides on a smooth horizontal plane.

7 Obtain the equations ofmotion of the system using lagrange s Determine the equations of motion of an insect of mass m crawling at a uni-form speedv on a uniform heavy rod of mass M and length 2a which isturning about a fixed end. Assume that at t= 0 the insect is at the middlepoint of the rod and it is crawling A uniform rod of mass M and length 2a is attached at one end by a cord oflength l to a fixed point. Calculate the inclination of the string and the rodwhen the string plus rod system revolves about the vertical through the pivotwith constant angular velocity . A particle moves in a horizontal plane in a central force potential U(r). Derivethe Lagrangian in terms of the polar coordinates (r, ).

8 Find the correspondingmomenta prand p and the Hamiltonian . Hence show that the energy andangular momentum of the particle are conserved.[University of Manchester 2007] Consider the system consisting of two identical masses that can move hori-zontally, joined with springs as shown in Let x, y be the horizontaldisplacements of the two masses from their equilibrium positions.(a) Find the kinetic and potential energies of the system and deduce theLagrangian.(b) Show that lagrange s equation gives the coupled linear differential equa-tions m x= 4k x+ 3k ym y= 3k x 4k yFig. Problems291(c) Find the normal modes of oscillation of this system and their period Two identical beads of mass m each can move without friction along a hor-izontal wire and are connected to a fixed wall with two identical springs ofspring constant k as shown in (a) Find the Lagrangian for this system and derive from it the equations ofmotion.

9 (b) Find the eigenfrequencies of small amplitude oscillations.(c) For each normal mode, sketch the system when it is at : Your sketch should indicate the relative sizes as well as the directions ofthe displacements.[University of Manchester 2007] Two beads of mass 2m and m can move without friction along a horizontalwire. They are connected to afixed wall with two springs of spring constants2k and k as shown in :(a) Find the Lagrangian for this system and derive from it the equations ofmotion for the beads.(b) Find the eigenfrequencies of small amplitude oscillations.(c) For each normal mode, sketch the system when it is at the maximum Lagrangian and Hamiltonian Three identical particles of mass m, M and m with M in the middle are con-nected by two identical massless springs with a spring constant k.

10 Find thenormal modes of oscillation and the associated (a) A bead of mass m is constrained to move under gravity along a planarrigid wire that has a parabolic shape y= x2/ l, where x and y are,respectively, the horizontal and the vertical coordinates. Show that theLagrangian for the system isL=m( x)22 1+4x2l2 mgx2l(b) Derive the Hamiltonian for a single particle of mass m moving in onedimension subject to a conservative force with a potential U(x).[University of Manchester 2006] A pendulum of length l and mass m is mounted on a block of mass M. Theblock can move freely without friction on a horizontal surface as shown (a) Show that the Lagrangian for the system isL= M+ m2 ( x)2+ ml cos x +m2l2( )2+ mgl cos (b) Show that the approximate form for this Lagrangian , which is applicablefor a small amplitude swinging of the pendulum, isL= M+ m2 ( x)2+ ml x +m2l2( )2+ mgl 1 22 (c) Find the equations of motion that follow from the simplified Lagrangianobtained in part (b),(d) Find the frequency of a small amplitude oscillation of the system.