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UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
APM3712
Mechanics and calculus of variations
Welcome to the APM3712 examination
Date:11 NOV 2025
Time: 8:00 - 10:30
Hours: 2 Hours and 30 Minutes
Examiner name: Prof H Jafari
Internal moderator name: Prof A. Kubeka
External moderator name: Prof. K. Patidar
This paper consists of 5 pages (including the cover page).
Total marks: 100 Marks
Number of pages: 5 pages Number of Questions: 5
Instructions:
• Include reference to additional information sheets if applicable.
• You are allowed to use a pocket and non-programming calculator.
• No typed solutions will be accepted and the script will be cancelled
• You need to declare your honesty regarding writing this paper.
Additional student Instructions:
1. Students must upload their answer scripts in a single PDF file (answer scripts must
not be password protected or uploaded as “read only” files)
2. Incorrect file format and uncollated answer scripts will not be considered.
3. NO emailed scripts will be accepted.
4. Students are advised to preview submissions (answer scripts) to ensure legibility and that
the correct answer script file has been uploaded.
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5. Incorrect answer scripts and/or submissions made on unofficial examinations platforms
(including the invigilator cell phone application) will not be marked and no opportunity will
be granted for resubmission. Only the last answer file uploaded within the stipulated
submission duration period will be marked.
6. Mark awarded for incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
7. Mark awarded for illegible scanned submission will be the student’s final mark. No
opportunity for resubmission will be granted.
8. Submissions will only be accepted from registered student accounts.
9. Students who have not utilised the proctoring tool will be deemed to have transgressed
Unisa’s examination rules and will have their marks withheld. If a student is found to have
been outside the proctoring tool for a total of 10 minutes during their examination session,
they will be considered to have violated Unisa’s examination rules and their marks will be
withheld. For examinations which use the IRIS invigilator system, IRIS must be recording
throughout the duration of the examination until the submission of the examinations scripts.
10. Students have 48 hours from the date of their examination to upload their invigilator results
from IRIS. Failure to do so will result in students deemed not to have utilized the proctoring
tools.
11. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. Students may not communicate with any other person or request
assistance from any other person during their examinations. Plagiarism is a violation of
academic integrity and students who plagiarise, copy from published work or Artificial
Intelligence Software (eg ChatGPT) or online sources (eg course material), will be in violation
of the Policy on Academic Integrity and the Student Disciplinary Code and may be referred
to a disciplinary hearing. Unisa has a zero tolerance for plagiarism and/or any other forms of
academic dishonesty.
12. Listening to audio (music) and making use of audio-to-text software is strictly prohibited
during your examination session unless such usage of the software is related to a student’s
assistive device which has been so declared. Failure to do so will be a transgression of
Unisa’s examination rules and the student's marks will be withheld.
13. Students are provided 30 minutes to submit their answer scripts after the official
examination time. Students who experience technical challenges should report the
challenges to the SCSC on 080 000 1870 or their College exam support centres (refer
to the Get help during the examinations by contacting the Student Communication Service
Centre [unisa.ac.za]) within 30 minutes. Queries received after 30 minutes of the official
assessment duration time will not be responded to. Submissions made after the official
assessment time will be rejected according to the examination regulations and will not be
marked. Only communication received from your myLife account will be considered.
14. Non-adherence to the processes for uploading assessment responses will not qualify the
student for any special concessions or future assessments.
15. Queries that are beyond Unisa’s control include the following: a. Personal network or
service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer
d. Non-functioning cameras or web cameras
e. Using work computers that block access to the myExams site (employer firewall
challenges)
f. Unlicensed software (eg license expires during exams)
Postgraduate students experiencing the above challenges are advised to apply for an
aegrotat and submit supporting evidence within ten days of the examination session.
Students will not be able to apply for an aegrotat for a third examination opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their second
examination opportunity will have to reregister for the affected module.
16. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. UNISA has a zero tolerance for plagiarism and/or any other forms
of academic dishonesty.
17. Students experiencing network or load shedding challenges are advised to apply together
with supporting evidence for an Aegrotat within 3 days of the examination session.
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QUESTION 1
Consider the variational problem with Lagrangian function
L(t, x, y, z, ẋ, ẏ, ż) = ẋ2 + ẏ2 − ż2 − ẏ + ż − 4 txyz.
(1.1) Obtain the Euler-Lagrange equations ( Do NOT solve them). (9)
(1.2) For this system
(a) Construct the Hamiltonian function from L. (6)
(b) Obtain the Hamilton’s equations. (3)
(c) Write down the Hamilton-Jacobi equation. (3)
[21]
QUESTION 2
(2.1) Consider the variational problem with Lagrangian function
L (t, x, ẋ) = ẋ2 − ẋ−2x sin t,
and endpoint conditions x(0) = 1, x( π2 ) = 2.
(a) Find the smooth extremal of the given variational problem. (9)
(b) Compare the value of the fundamental integral (8)
Z
L (t, x, ẋ) dt
along an extremal between those two endpoints with the value along the line x(t) = cos t through
the two
R points.
Hint: sin2 t dt = 2t − 14 sin(2t), cos2 t dt = 2t + 14 sin(2t), 2 sin t cos t dt = − 21 cos(2t).
R R
(2.2) Use the Weierstrass Excess function to determine the nature of the extremal found in (2.1(a)). (6)
[23]
[TURN OVER]
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QUESTION 3
(3.1) We know that for L = L(t, x, ẋ), the Euler-Lagrange equation is calculated by
d ∂L ∂L
− = 0.
dt ∂ ẋ ∂x
∂L
(a) Show that the if L is independent of x, (it means ∂x = 0), then the Euler-Lagrange equation gives (4)
∂L
= c, where c is a constant.
∂ ẋ
(b) Use part (3.1(a)), to find the smooth extremal of the following variational problem. (4)
ẋ2
L(t, ẋ) = .
cos t
(3.2) Obtain the extremals (if they exist) of the following problem of Lagrange with functional (12)
Z 1
ẋ2 + 2ty + 2x dt,
I[X] =
0
subject to the following auxiliary and boundary conditions:
ẋ + ẏ = 0,
x(0) = 1, y(0) = 0,
4 1
x(1) = , y(1) = − .
3 3
[20]
QUESTION 4
(4.1) Find the Euler-Lagrange equation for the variational problem with the fundamental integral (9)
Z z2 Z y2 Z x2
yz U2x + 5 U2y + xz2 U2z + 4xyz U2 dx dy dz,
z1 y1 x1
∂U ∂U ∂U
where U = U(x, y, z) and Ux = , Uy = , Uz = .
∂x ∂y ∂z
(4.2) Consider the integral
Z t2
1 2
I= (ẋ + ẏ2 )dt,
t1 2
and the 1−parameter transformation
x̄ = x + α y, ȳ = y − α x, t̄ = (1 + α2 ) t.
(a) Show that the given integral I is invariant under the given 1−parameter transformation. (7)
(b) Use Noether’s theorem and part (4.2(a)) to show that xẏ − yẋ is constant along an extremal of (6)
the Lagrangian
1
L(ẋ, ẏ) = (ẋ2 + ẏ2 ).
2
[22]
[TURN OVER]
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UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
APM3712
Mechanics and calculus of variations
Welcome to the APM3712 examination
Date:11 NOV 2025
Time: 8:00 - 10:30
Hours: 2 Hours and 30 Minutes
Examiner name: Prof H Jafari
Internal moderator name: Prof A. Kubeka
External moderator name: Prof. K. Patidar
This paper consists of 5 pages (including the cover page).
Total marks: 100 Marks
Number of pages: 5 pages Number of Questions: 5
Instructions:
• Include reference to additional information sheets if applicable.
• You are allowed to use a pocket and non-programming calculator.
• No typed solutions will be accepted and the script will be cancelled
• You need to declare your honesty regarding writing this paper.
Additional student Instructions:
1. Students must upload their answer scripts in a single PDF file (answer scripts must
not be password protected or uploaded as “read only” files)
2. Incorrect file format and uncollated answer scripts will not be considered.
3. NO emailed scripts will be accepted.
4. Students are advised to preview submissions (answer scripts) to ensure legibility and that
the correct answer script file has been uploaded.
Downloaded by Stephen ()
, lOMoARcPSD|54878315
5. Incorrect answer scripts and/or submissions made on unofficial examinations platforms
(including the invigilator cell phone application) will not be marked and no opportunity will
be granted for resubmission. Only the last answer file uploaded within the stipulated
submission duration period will be marked.
6. Mark awarded for incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
7. Mark awarded for illegible scanned submission will be the student’s final mark. No
opportunity for resubmission will be granted.
8. Submissions will only be accepted from registered student accounts.
9. Students who have not utilised the proctoring tool will be deemed to have transgressed
Unisa’s examination rules and will have their marks withheld. If a student is found to have
been outside the proctoring tool for a total of 10 minutes during their examination session,
they will be considered to have violated Unisa’s examination rules and their marks will be
withheld. For examinations which use the IRIS invigilator system, IRIS must be recording
throughout the duration of the examination until the submission of the examinations scripts.
10. Students have 48 hours from the date of their examination to upload their invigilator results
from IRIS. Failure to do so will result in students deemed not to have utilized the proctoring
tools.
11. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. Students may not communicate with any other person or request
assistance from any other person during their examinations. Plagiarism is a violation of
academic integrity and students who plagiarise, copy from published work or Artificial
Intelligence Software (eg ChatGPT) or online sources (eg course material), will be in violation
of the Policy on Academic Integrity and the Student Disciplinary Code and may be referred
to a disciplinary hearing. Unisa has a zero tolerance for plagiarism and/or any other forms of
academic dishonesty.
12. Listening to audio (music) and making use of audio-to-text software is strictly prohibited
during your examination session unless such usage of the software is related to a student’s
assistive device which has been so declared. Failure to do so will be a transgression of
Unisa’s examination rules and the student's marks will be withheld.
13. Students are provided 30 minutes to submit their answer scripts after the official
examination time. Students who experience technical challenges should report the
challenges to the SCSC on 080 000 1870 or their College exam support centres (refer
to the Get help during the examinations by contacting the Student Communication Service
Centre [unisa.ac.za]) within 30 minutes. Queries received after 30 minutes of the official
assessment duration time will not be responded to. Submissions made after the official
assessment time will be rejected according to the examination regulations and will not be
marked. Only communication received from your myLife account will be considered.
14. Non-adherence to the processes for uploading assessment responses will not qualify the
student for any special concessions or future assessments.
15. Queries that are beyond Unisa’s control include the following: a. Personal network or
service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer
d. Non-functioning cameras or web cameras
e. Using work computers that block access to the myExams site (employer firewall
challenges)
f. Unlicensed software (eg license expires during exams)
Postgraduate students experiencing the above challenges are advised to apply for an
aegrotat and submit supporting evidence within ten days of the examination session.
Students will not be able to apply for an aegrotat for a third examination opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their second
examination opportunity will have to reregister for the affected module.
16. Students suspected of dishonest conduct during the examinations will be subjected to
disciplinary processes. UNISA has a zero tolerance for plagiarism and/or any other forms
of academic dishonesty.
17. Students experiencing network or load shedding challenges are advised to apply together
with supporting evidence for an Aegrotat within 3 days of the examination session.
Downloaded by Stephen ()
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QUESTION 1
Consider the variational problem with Lagrangian function
L(t, x, y, z, ẋ, ẏ, ż) = ẋ2 + ẏ2 − ż2 − ẏ + ż − 4 txyz.
(1.1) Obtain the Euler-Lagrange equations ( Do NOT solve them). (9)
(1.2) For this system
(a) Construct the Hamiltonian function from L. (6)
(b) Obtain the Hamilton’s equations. (3)
(c) Write down the Hamilton-Jacobi equation. (3)
[21]
QUESTION 2
(2.1) Consider the variational problem with Lagrangian function
L (t, x, ẋ) = ẋ2 − ẋ−2x sin t,
and endpoint conditions x(0) = 1, x( π2 ) = 2.
(a) Find the smooth extremal of the given variational problem. (9)
(b) Compare the value of the fundamental integral (8)
Z
L (t, x, ẋ) dt
along an extremal between those two endpoints with the value along the line x(t) = cos t through
the two
R points.
Hint: sin2 t dt = 2t − 14 sin(2t), cos2 t dt = 2t + 14 sin(2t), 2 sin t cos t dt = − 21 cos(2t).
R R
(2.2) Use the Weierstrass Excess function to determine the nature of the extremal found in (2.1(a)). (6)
[23]
[TURN OVER]
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QUESTION 3
(3.1) We know that for L = L(t, x, ẋ), the Euler-Lagrange equation is calculated by
d ∂L ∂L
− = 0.
dt ∂ ẋ ∂x
∂L
(a) Show that the if L is independent of x, (it means ∂x = 0), then the Euler-Lagrange equation gives (4)
∂L
= c, where c is a constant.
∂ ẋ
(b) Use part (3.1(a)), to find the smooth extremal of the following variational problem. (4)
ẋ2
L(t, ẋ) = .
cos t
(3.2) Obtain the extremals (if they exist) of the following problem of Lagrange with functional (12)
Z 1
ẋ2 + 2ty + 2x dt,
I[X] =
0
subject to the following auxiliary and boundary conditions:
ẋ + ẏ = 0,
x(0) = 1, y(0) = 0,
4 1
x(1) = , y(1) = − .
3 3
[20]
QUESTION 4
(4.1) Find the Euler-Lagrange equation for the variational problem with the fundamental integral (9)
Z z2 Z y2 Z x2
yz U2x + 5 U2y + xz2 U2z + 4xyz U2 dx dy dz,
z1 y1 x1
∂U ∂U ∂U
where U = U(x, y, z) and Ux = , Uy = , Uz = .
∂x ∂y ∂z
(4.2) Consider the integral
Z t2
1 2
I= (ẋ + ẏ2 )dt,
t1 2
and the 1−parameter transformation
x̄ = x + α y, ȳ = y − α x, t̄ = (1 + α2 ) t.
(a) Show that the given integral I is invariant under the given 1−parameter transformation. (7)
(b) Use Noether’s theorem and part (4.2(a)) to show that xẏ − yẋ is constant along an extremal of (6)
the Lagrangian
1
L(ẋ, ẏ) = (ẋ2 + ẏ2 ).
2
[22]
[TURN OVER]
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