Special Relativity and Classical Field Theory · 2020-05-22 · Special Relativity and Classical...

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Special Relativity and Classical Field Theory Lecture and Tutorial – Prof. Dr. Haye Hinrichsen – B.Sc. Moritz Dorband – SS 2020 Without internet, Emmy Noether discussed abstract algebra with her friends via postcards. Exercise 5.1: Polynomial action (3P) In the lecture we derived the action of a point particle with mass m in a potential V : S [x, ˙ x]= Z τ B τ A -mc p -η μν ˙ x μ ˙ x ν + V (x) dτ. Some theorists do not like this action because of the nasty square root. They would rather prefer a polynomial action. In this exercise let us study the following polynomial action S [x, ˙ x]= Z τ B τ A L(x, ˙ x) dτ = Z τ B τ A 1 2ξ ˙ x μ ˙ x μ - ξm 2 c 2 2 + V (x) dτ, where ξ (τ ) is an additional independent function. (a) Compute the variation δS =0 with respect to δx μ and δξ and derive the corre- sponding equations of motion. (2P) (b) Insert the solution for ξ (τ ) into the other equation of motion and show that we get the same results as for the ordinary action (see lecture notes). (1P) Exercise 5.2: Center of mass theorem (6P) Consider N free non-relativistic particles with different masses m n and coordinates q n , ˙ q n in one dimension, as described by the Lagrange function L = 1 2 N n=1 m n ˙ q 2 n . The aim of this exercise is to compute the Noether charge corresponding to a Galilei transformation q n q (s) n = q n - svt where s is an infinitesimal parameter. (a) Confirm that the Galilei transformation is a symmetry transformation of L. (2P) (b) Show that the Noether charge is given by (1P) Q = v N X n=1 m n ( q n - ˙ q n t). Exercises Sheet 5 Please turn over

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Page 1: Special Relativity and Classical Field Theory · 2020-05-22 · Special Relativity and Classical Field Theory LectureandTutorial–Prof.Dr.HayeHinrichsen–B.Sc.MoritzDorband–SS2020

Special Relativity and Classical Field TheoryLecture and Tutorial – Prof. Dr. Haye Hinrichsen – B.Sc. Moritz Dorband – SS 2020

Without internet, Emmy Noether discussed abstract algebra with her friends via postcards.

Exercise 5.1: Polynomial action (3P)

In the lecture we derived the action of a point particle with mass m in a potential V :

S[x, x] =

∫ τB

τA

(−mc

√−ηµν xµxν + V (x)

)dτ.

Some theorists do not like this action because of the nasty square root. They would ratherprefer a polynomial action. In this exercise let us study the following polynomial action

S[x, x, ξ] =

∫ τB

τA

L(x, x, ξ) dτ =

∫ τB

τA

( 1

2ξxµx

µ − ξm2c2

2+ V (x)

)dτ,

where ξ(τ) is an additional independent function.

(a) Compute the variation δS = 0 with respect to δxµ and δξ and derive the corre-sponding equations of motion. (2P)

(b) Insert the solution for ξ(τ) into the other equation of motion and show that we getthe same results as for the ordinary action (see lecture notes). (1P)

Exercise 5.2: Center of mass theorem (6P)

Consider N free non-relativistic particles with different masses mn and coordinates qn, qnin one dimension, as described by the Lagrange function L = 1

2

∑Nn=1mnq

2n. The aim of

this exercise is to compute the Noether charge corresponding to a Galilei transformation

qn → q(s)n = qn − svt

where s is an infinitesimal parameter.

(a) Confirm that the Galilei transformation is a symmetry transformation of L. (2P)

(b) Show that the Noether charge is given by (1P)

Q = v

N∑n=1

mn

(qn − qnt).

Exercises Sheet 5Please turn over ⇒

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(c) Explain why the conservation of Q tells us that the center of mass

q(t) =

∑Nn=1mnqn(t)∑N

n=1mn

of the system moves uniformly. (2P)

(d) Suppose that we add a harmonic potential between the particles of the form

L =1

2

N∑n=1

mnq2n −

1

2

N∑m,n=1

(qn − qm)2.

What changes in (a)-(c)? (1P)

Exercise 5.3: Noether charges under Lorentz transformations (3P)

Consider a relativistic particle described by a Lorentz-invariant Lagrange function L(x, x).

(a) Show that for an infinitesimal transformation generated by the SO+(3, 1) generatorλ(αβ) (see lecture notes) the corresponding conserved Noether charge is given by(2P)

Qαβ = pαxβ − pβxα.

(b) Give an interpretation of the components of Qαβ. (1P)

(Σ = 12P)

Please submit your solution as a single pdf file via WueCampus. Deadline is Friday, May 29 at 12:00.

Exercises Sheet 5