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Lagrangian Mechanics For Dummies: An Intuitive Introduction – Profound Physics
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Write the Lagrangian discussed in Eq (6.27) to (6.30) in polar coordinates for the case V11-V22>0 and V12-V21=0 . Show that there is a radial normal mode r r0 cos(a r) with frequency ω=(V11)^1 / 2 when the angular momentum is zero. Show that in the case of nonzero angular momentum, the angular momentum is conserved and the particle can no longer reach r=0 . Write the fictitious potential energy V^'(r) (Chapter 3 ) for nonzero angular momentum. When finished, reintroduce the mass parameter, m, in
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SOLVED: A relativistic point particle has a Lagrangian(3)where c is the speed of light. Derive the Euler-Lagrange equations. Show that they reduce to Newton's equation in the limit2(4)In the case that V=0, a free relativistic particle, show thatt=a t(5)is still a solution for constant vectors a and (of course we need c^2 > 0 for L to be real).
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