solutions_HW5

# solutions_HW5 - Homework 5 Classical Mechanics (Phys701)...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 5 Classical Mechanics (Phys701) Due: October 10, 2011 Problem 1-7 Show that Lagrange’s equations can also be written as ∂ ˙ T ∂ ˙ q j − 2 ∂T ∂q j = Q j . These are sometimes known as the Nielsen form of the Lagrange equations. (Hint: start with writing a formula for ˙ T ). Solution : First, calculate d dt T ( ˙ q i , q i , t ) = ∂T ∂ ˙ q i ¨ q i + ∂T ∂q i ˙ q i + ∂T ∂t . Consequently ∂ ∂ ˙ q j dT dt = ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂T ∂q j + ∂ 2 T ∂ ˙ q j ∂t . Then the Nielsen equation reads: ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂ 2 T ∂ ˙ q j ∂t − ∂T ∂q j = 0 . However, the first three terms here can be identified as ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂ 2 T ∂ ˙ q j ∂t = d dt ∂T ∂ ˙ q j (check it directly) and we get d dt ∂T ∂ ˙ q j − ∂T ∂q j = 0 , i.e., the Lagrange equation. Problem 1-9 The electromagnetic field is invariant under a gauge transformation of the scalar and vector potentials given by A → A + ∇ Ψ( r, t ) , ϕ → ϕ − 1 c ∂ Ψ ∂t , where Ψ is an arbitrary (but differentiable). What effect does this gauge transformation have on the Lagrangian of a particle moving in the electromagnetic field? Is the motion affected? Solution . 1 v eff + r _ r E eff Figure 1: Dependence V eff ( r ) and the graphic solution for the values of r ± . Problem 1-21 Two mass points of mass m 1 and m 2 are connected by a string passing through a hole in a smooth table so that m 1 rests on the table surface and m 2 hangs suspended. Assuming m 2 moves only only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for the systemline, what are the generalized coordinates for the system?...
View Full Document

## This note was uploaded on 12/13/2011 for the course PHYS 701 taught by Professor Bazaliy during the Fall '11 term at South Carolina.

### Page1 / 4

solutions_HW5 - Homework 5 Classical Mechanics (Phys701)...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online