Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Midterm 3 Exam for Phys601: Covariance and Transformations in Physics, Exams of Physics

University of MarylandPhysics

A take-home midterm exam for a university-level physics course, phys601. The exam covers topics such as covariance, transformations, and relativistic mechanics. Students are expected to work independently and may consult written materials. The exam includes five problems, each worth a certain number of points. The first problem deals with the covariance of a given equation under galilean transformations. The second problem involves finding the 4-canonical momentum and hamiltonian for a charge in an electromagnetic field. The third problem deals with the transformation of phase space coordinates and finding the hamilton equations. The fourth problem involves finding the period of motion for a particle under a potential using action-angle variables. The fifth problem deals with a mass moving vertically in the earth's gravitational field using hamilton-jacobi theory.

Typology: Exams

Pre 2010

Uploaded on 07/30/2009

koofers-user-z53
koofers-user-z53 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Phys601/F08/Midterm 3
Take-Home Out 12/16/08 4PM
Return to my office 12/19/08 < 12Noon (slip under door; or email copy is fine)
• The test tests concepts, so significant points will be taken off for conceptual errors
• Please box all important answers else, things can be missed easily.
• Be concise More words does not translate to partial credit (see above).
• You are expected to work independently. You may consult any written material.
• Can ask me clarification questions by email.
Problem 1 (20 points)
A new equation of nature is discovered. In the lab frame (S), the equation of a field ψ(x,t) is
given as ∂ψ/∂t + c ∂ψ/∂x = 0. The discoverer insists that the speed c is a universal constant, ie, it
must be the same in all inertial frames of reference. (x,t) are spacetime coordinates.
1. Show that under a Galilean transformation to a frame S’ moving with V < c, the equation
transformed to (x’,t’) coordinates is not covariant if c is a universal constant.
2. Consider a more general transformation of the form x’k = Aklxl, where xk → (x,ct). By
demanding covariance, find Akl, the transformation matrix A(V), assuming that the
matrix for the inverse transformation from S’ to S must be just A(-V).
[Assume that the equation given is true only for V < c and ignore the fact that the solutions to
this equation have a preferred direction.]
Problem 2 (10 points)
In a homework problem (Set 11, solutions posted), we found the relativistic Lagrangian for a
charge q mass m in an electromagnetic field, the latter specified in terms of the 4-vector Aμ.
1. What is the 4-canonical momentum Pμ.
2. Find the Hamiltonian in manifestly covariant notation.
[Co and contravariant indices must be correctly used for credit for this problem.]
pf2

Partial preview of the text

Download Midterm 3 Exam for Phys601: Covariance and Transformations in Physics and more Exams Physics in PDF only on Docsity!

Phys601/F08/Midterm 3

Take-Home Out 12/16/08 4PM Return to my office 12/19/08 < 12Noon (slip under door; or email copy is fine)

  • The test tests concepts, so significant points will be taken off for conceptual errors
  • Please box all important answers else, things can be missed easily.
  • Be concise More words does not translate to partial credit (see above).
  • You are expected to work independently. You may consult any written material.
  • Can ask me clarification questions by email.

Problem 1 (20 points)

A new equation of nature is discovered. In the lab frame (S), the equation of a field ψ(x,t) is given as ∂ψ/∂t + c ∂ψ/∂x = 0. The discoverer insists that the speed c is a universal constant, ie, it must be the same in all inertial frames of reference. (x,t) are spacetime coordinates.

  1. Show that under a Galilean transformation to a frame S’ moving with V < c, the equation transformed to (x’,t’) coordinates is not covariant if c is a universal constant.
  2. Consider a more general transformation of the form x’k = Akl xl , where x (^) k → (x,ct). By demanding covariance, find Akl, the transformation matrix A(V), assuming that the matrix for the inverse transformation from S’ to S must be just A(-V).

[Assume that the equation given is true only for V < c and ignore the fact that the solutions to this equation have a preferred direction.]

Problem 2 (10 points)

In a homework problem (Set 11, solutions posted), we found the relativistic Lagrangian for a charge q mass m in an electromagnetic field, the latter specified in terms of the 4-vector Aμ.

  1. What is the 4-canonical momentum Pμ.
  2. Find the Hamiltonian in manifestly covariant notation.

[Co and contravariant indices must be correctly used for credit for this problem.]

Problem 3 (20 points)

We are given a transformation in phase space specified by the relations P=P(p,q), Q=Q(p,q). There is no explicit time dependence in the transformation. A Hamiltonian H(p,q) is also given.

  1. Find an expression for dQ/dt involving only partial derivatives of P, Q, and H with respect to {p,q}.
  2. Considering that H is an implicit function of {P,Q}, find expressions for ∂H/∂p and ∂H/∂q in terms of ∂H/∂P and ∂H/∂Q and partial derivatives of {P, Q} with respect to {p,q}.
  3. If the P and Q transformations are canonical, what are the Hamilton equations in the {P,Q} system? By using 1 and 2 above, find a necessary condition involving only partials of {P,Q} with respect to {p,q}. [There is a parallel condition to the above, obtainable by computing dP/dt. You don’t have to do it.]
  4. Test whether the condition in 3 applies to the transformations given in Goldstein 9.4.

Problem 4 (20 points) (Goldstein 10.13)

A particle of mass m moves in periodic motion in one dimension under the influence of a potential V = F|x|, where F is a constant. Using action-angle variables, find the period of the motion as a function of the particle's energy, E.

Problem 5 (20 points)

Consider a mass m moving vertically in the Earths gravitational field, g. The x-coordinate increases upward, g points down.

  1. Write down the Hamiltonian. Rewrite this in normalized {p,q} coordinates, using a height h to normalize the x coordinate.
  2. Use Hamilton-Jacobi theory to solve for p(t) and q(t) in terms of constants α and β where, for the P coordinate, we pick P = H and that P(0) = α. β is another constant of the motion such that Q(0) = β.
  3. Write down also expressions for the transformed Hamilton-Jacobi coordinates α(p,q,t) and β(p,q,t). Make two contour plot sketches of the {α,β} grid at t=0 and at some other t > 0. Depict how a test particle phase space point that starts at {q=0, p > 0} moves with the grid.