Question

# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends on r alone. Also, show that the motion of a particle is confined onto a two-dimensional plane. Q2) Using polar coordinates (r, theta) in this plane, show a Lagrangian, 'L', for the motion of a particle. Also, show a Hamiltonian, 'H'. Q3) Show an equation of motion in both 'r' and 'theta' directions, respectively. Q4) When the orbit of a particle is exactly circular, show the equation for its radius. Q5) Suppose that U(r) is given as, U(r) = k r**2/2, where k> 0 and 'r**2' denotes r squared. Suppose also that an angular velocity of a circular orbit in this potential is denoted as 'Omega = d theta / dt' (differential of theta with time). Now suppose that the motion of a particle deviates slightly from this circular orbit, thereby showing a radial oscillation around this orbit with a specific frequency. Show this frequency using 'Omega'.

Answer 1

When the potential energy function is a function of the scalar distance r=|vec r| from a fixed position in space (centre of force) we call this potential energy function is centrally symmetric and the corresponding force field is 'central'.
Problem 1: act ur) us consider where r is a the particle of mass m position vector of moving in a any arbitrary gravitational pote pourt from the of potential. 8 any arbitrary pourt from the center In case where the potential energy function ů a function of r a we say that the potential energy function ý centrally symmetric an corresponding force field is central. f(a) = - 2U (r) ar r.e. the force is always directed towards or away from the centre (origin). 1 Since the central force has only a dependence, it implies that the isotropy. of space is conserved about its origin; i, e. the total angular momentrum of the system about the origin i conserved. Now, torque, t = 5x f(F) = 0 - E = dl =0 Again, I = ?= constant ; I x P of motion fixed out in space; defined by velocity article moves in planar. Now, the central force i ýalways directed towards a so it has no component perpendicular to the plane rector and the direction of the force. Hence the flane only. This motion under central force is ?=mrxe = conotant i ř and ů lie in a plane perpendicular to the fixed vector I. 2) The Lagrangian of a Here, L= 1 m ( particle ů given by L=T-V + r28²) – U(r) ; the motion of the particle is confined onto a two-dimensional plane. Hoca, Tog moment to mee) -4m (it si + sr* ainºo $2) 0 = 0, T= m ( +²6²)

Now, Hami Hamiltonian, H = b. , -L = ribe + 8 Pe - Bar a mi > * = Parent ; po = 24 - medo > - bem H= pre be + Po hem to Po - Por am + m Pe L met roker m4 / + Ha bu + b + u(x) 4 2m 3) det x= rcose, y = Asino er en de cose - osime den --- en het e meno + recoso de ... e fruits - ( Jeso - Convento + 2 dien] ano... ] kio + [ v. de 2 de dese] caso ... Les sleceller ation, ă a se face - Le (icos e + j dur e) - ( si eu Acceleration, a fruit-ting) Now, i = (i cose ti sino); ô = (-isine ti coso) să = Cian + ông) & - (nu det O d2 Op = for den na die dorp --(0) According to Newton's second law of motion, f(x) = f(+) 5 f(x) = = m(ta, + 6a) (1)

m computing. m plante me mos) ] = 40.. and minden a dit ] = 0...19 of motion of the particle moving Equer (12) & (13) under a central are force called equation field. ar mr3 ar 4 For circular orbit, off (7) = 1 + 0(2) OU (W) - - + Qua) -o » mue" com - JUC) ; : L- mico we angular velocity mrwn = aver + hot sau(a) = fade with the mai (2) => r- 2U(x) + c which is the reqd, egun for the radius of the circular orbit. I mw?