Question

Keppler's problem - motion of a planet in the orbit around the Sun, for problems 2 and 3. The potential of interaction is gravitational: V=-k/r where k is constant. Problem 3 Use Lagrange's formalism to find the equations of motion. Find the first integral and indicate which quantities are conserved, Problem 4 Use Hamiltons formalism to find the same equations of motion. Find the first integral and indicate which quantities are conserved,

Answer 1

To obtain equation of motion by a lagrargian formalism we use polar coordinate cr, o Now kinetic Energy to emer2+82 0²) - and potential energy V = VN) =- - * Now Lagrangian Larv La LM (42+7202) -ven) Now equation of motion of fat ( 22 ) - 2 2 2 2 20 9 from egn ② 21 20 and 21 uur from egr a 1 (2026)=0 T m720-0 = Constant = 3 57 where I is first Integral represent angular momentim magnitude . as mis constant, egni giny d (828): 0 6 1920 = Constant 2 careal velocity] thus dla k o From egn ② 2 2 2 2 2 mi 37 / 2 mr av

mr2 Now Lagrangian egn of ( 22 ) - 2 / 20 Now (mm) -476?+34 20 Now FC = - egne må-mro 2= For) - @ - This is second order differential egn we 0 I (From 5) - To Imer2+8282] = 1 mo2t 2 3 4 Potential energy v= -2 To tal Energy E = T+V= 4 manat arat (5) * - Constant the total Energy of the system in Constant This is another First Integral of motion egn 9 reloresents equation of motion while angular momentum and total energy are Constant of motion, This quantities are Areal velocity, angular momentum and total energy Conserned. are

o Now Hamiltonian Ma E Prix-2 (11) Lagrangian 2 = Tv = Imer2+r202) + k do Pra mo 12 m m 2m 9m72 r ad 1 From @ Hi Prit Poo -2 H = Partpoo - Slme2?1726 17 ] Hz P4 48068-4m*?-?m=282 Ha by² + Po? - P² P2 k Thus H2 82 + b 2 - 1 - egr 1 gines 24 l = 2 2 4 2 - p Hamilton sen Lo - Se - Home page 3 - hot and Now From abone egry we get Str = o - HOP + 2 = - am 12 m2

As premi From so por = m% @ @ and @ we get -mř= - po? mrz+ - mr=-(moro)? mygth 7?m= h = radach But 1. How let u= do yo £ 2. lo do o lhua - oe hur a-hody 02 1 from egr 1 Tur planetsy merioๆ % z_ heue oru 22 20 and 2 heue deu a = k hut - 1 au 2 au - + U dos m3

Thury I we know ahide=0 1. But 29120=-Po Pozo 3 d CPol=0 Po = Constant a mazo= Constant (since in constant) or red' = Constant ? J CFirst Integral) (this areal velocity's Constant J= 1 moze 7 Same as part ca) quantities Conserved areal velocity angular momentum, tosal Energy