APM3712 Assignment 3 2026 Year Module Solutions Due 2026

UNIVERSITY OF SOUTH AFRICA
Department of Mathematical Sciences


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APM3712: Mechanics
and Calculus of Variations
Assignment 03 — Year Module

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APM3712
Module Code:
Mechanics and Calculus of Variations
Module Name:
Department of Mathematical Sciences
Department:
Assignment 03
Assignment Number:
Year Module
Module Type:




Submitted in partial fulfilment of the requirements
for Mechanics and Calculus of Variations — UNISA

,UNISA | APM3712 Mechanics and Calculus of Variations — Assignment 03



Q1

Construct the Hamiltonian function, derive the canonical equations and write down the Hamilton–
Jacobi equation for the following Lagrangian function


L(t, x, y, z, ẋ, ẏ, ż) = ẋ2 + ẏ 2 − ż 2 − 4xy + z.



A:

The generalized momenta are

∂L px
px = px = 2ẋ ẋ =
∂ ẋ 2


The generalized momentum corresponding to y is

∂L py
py = py = 2ẏ ẏ =
∂ ẏ 2


The generalized momentum corresponding to z is

∂L pz
pz = pz = −2ż ż = −
∂ ż 2


The Hamiltonian is
H = px ẋ + py ẏ + pz ż − L


Substitute the velocities

p 
x
p 
y
 p 
z p2x p2y p2
H = px + py + pz − −L H= + − z −L
2 2 2 2 2 2


Now substitute the Lagrangian

 p 2
x
 p 2
y
 p 2
z p2x p2y p2
L= + − − − 4xy + z L= + − z − 4xy + z
2 2 2 4 4 4


Hence
!
p2x p2y p2 p2x p2y p2 p2x p2y p2
H= + − z − + − z − 4xy + z H= + − z + 4xy − z
2 2 2 4 4 4 4 4 4


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, UNISA | APM3712 Mechanics and Calculus of Variations — Assignment 03


The canonical equations are
∂H px
ẋ = ẋ =
∂px 2
∂H py
ẏ = ẏ =
∂py 2
∂H pz
ż = ż = −
∂pz 2

Also,

∂H ∂H ∂H
ṗx = − ṗx = −4y ṗy = − ṗy = −4x ṗz = − ṗz = 1
∂x ∂y ∂z


The Hamilton–Jacobi equation is
 
∂S ∂S ∂S ∂S
+ H x, y, z, , , =0
∂t ∂x ∂y ∂z


Substitute the Hamiltonian
 2  2  2
∂S 1 ∂S 1 ∂S 1 ∂S
+ + − + 4xy − z = 0
∂t 4 ∂x 4 ∂y 4 ∂z




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