Examples etc

Question 1

What does is meant by homogeneity of space?

Answer 1

The motion is independent of position (i.e. no v(r))

Question 2

How can we prove that space is homogeneous?

Answer 2

• Displacement of pos by 𝛿r, so r1’ = r1 + 𝛿r, and v1’ = v1
• New L’ = L + 𝛿L
• If space is homogeneous, then action is the same, 𝛿L = 0

Question 3

How do we express the Lagrangian after it has been displaced by 𝛿r?

Answer 3

L(r+𝛿r, v) = L(r, v) + dL/dx 𝛿x + dL/dy 𝛿y + dL/dz 𝛿z, so 𝛿L = L(r+𝛿r, v) - L(r,v) = ∇L.𝛿r

Question 4

What is ∇L equal to if 𝛿L = 0 for and 𝛿r?

Answer 4

∇L = 0, so dL/dqi = 0

Question 5

What do we find from the E-L equation from dL/dqi = 0?

Answer 5

d/dt(dL/dq’i) = dL/dqi = 0, so d/dt(pi) = 0, where pi is the generalised momentum -> pi is constant if space is homogeneous

Question 6

How can we express 𝛿L for many particles?

Answer 6

𝛿L = sum over a of dL/dra . 𝛿r, where a is the index of the particle

Question 7

If 𝛿L = 0 for any 𝛿r, what do we get the sum of dL/dra is equal to?

Answer 7

sum over a of dL/dra = 0

Question 8

What can we do with the E-L equation for many particles?

Answer 8

Sum all E-L equations: sum over a of dL/dra - sum over a of d/dt(dL/dr’a) = 0, so sum over a of d/dt(dL/dr’a) = 0

Question 9

What do we find from the sum of the E-L equations?

Answer 9

p(a) = dL/dr’a, so total momentum P = sum over a of p(a) -> P is conserved if space is homogeneous

Question 10

What is the one exception for the homogeneity of space with velocity depending on r?

Answer 10

Can hold if there are interactions between particles i.e. v(r2-r1)

Question 11

What is the conservation law for the momenta?

Answer 11

Any canonical momenta pi whose conjugate coordinate qi does not appear in L is conserved.

Question 12

What is meant by the homogeneity of time?

Answer 12

If L is not an explicit function of time, it doesn’t matter what time motion occurs.

Question 13

What can we infer from the Homogeneity of time?

Answer 13

If time is homogeneous, L = L(qi, qi’), so dH/dt = 0, H is conserved (E is conserved)

Question 14

What is meant by the isotropy of space?

Answer 14

Mechanical properties don’t change on rotating the system if space is isotropic. Conversely, special direction effects motion.

Question 15

What do we do to show the isotropy of space?

Answer 15

• Rotate space around z, define rotation as by 𝛿Ф = dФ z(hat), and change in position 𝛿r = r*sinθ 𝛿Ф Ф(hat)
• 𝛿r = 𝛿Ф X r, and want 𝛿v
• 𝛿v = d/dt(𝛿r) = 𝛿Ф X r’ = 𝛿Ф X v

Question 16

What is the change in Lagrangian for a single particle when rotation the space?

Answer 16

𝛿L = sum over i of (dL/dqi * 𝛿qi + dL/dqi’ *𝛿qi’) = dL/dr 𝛿r + dL/dv 𝛿v

Question 17

What is another way of writing dL/dqi and how can we write this?

Answer 17

Fi = ρi’ = dL/dqi, so 𝛿L = ρi’ . 𝛿r + ρi . 𝛿v

Question 18

What do we then substitute into 𝛿L?

Answer 18

• 𝛿r = 𝛿Ф Xr and 𝛿v = 𝛿Ф X v

- Can then rearrrange using law to 𝛿L = 𝛿Ф.(rXρ’ + vXρ) = d/dt(rXρ)

Question 19

What do we finally find from 𝛿L?

Answer 19

• 𝛿L = 𝛿Ф.d/dt(rXρ) = 0 if space isotropic
• For all 𝛿Ф, d/dt(rXρ) = 0 or d/dt(L) = 0, where L is angular momentum
• Isotropy of space means angular momentum is conserved

Question 20

What is the change in Lagrangian for many particles when rotation the space?

Answer 20

𝛿L = sum over a of (dL/dr(a) . 𝛿r(a) + dL/dv(a) . 𝛿v(a) = 0, where a is the particle index and 𝛿r(a) = 𝛿ФXr(a), and same of 𝛿v(a)

Question 21

What is the next step after finding the in Lagrangian for multiple particles?

Answer 21

• Use ρ(a) = dL/dv(a) and F(a) = dL/dr(a) = ρ’(a), and sub this into 𝛿L
• Sub in the values for 𝛿r(a) and 𝛿v(a) aswell

Question 22

What is the final version of 𝛿L for many particles?

Answer 22

𝛿L = 𝛿Ф*(sum over a of d/dt(r(a)Xρ(a)) = 0

Question 23

What do we find from the final version of 𝛿L for many particles?

Answer 23

sum over a of d/dt(r(a)Xρ(a)) = 0 so total angular momentum of system is conserved if space is isotropic

Question 24

What can we summarise from these Lagrangian rearrangements?

Answer 24

With n generlised coordinates, have up to 2n+1 conserved quantities (n linear momenta, n angular momenta and energy)

Question 25

What is Noether’s theorem?

Answer 25

If a symmetry in the Lagrangian exists, there is a corresponding constant of the motion.

Question 26

What is a good example to show this theory?

Answer 26

-Field is an infinite homogeneous plane, e.g. V constant at height z i.i. V(z), but T(x’, y’, z’)

Question 27

How can we find the conserved quantities in this example?

Answer 27

• Energy conserved as no t in L = t-V
• Linear momenta ρx and ρy conserved as L invariant to translation in x-y plane, not conserved in ρz
• angular momenta Lz conserved and L is invariant under rotation about z(hat)

Question 28

How can we find the conserved quantities for a field due to 2 point objects?

Answer 28

Since V is symmetric on rotation around x, Lx is conserved, and energy is conserved, but everything else is not. Component of angular momentum along an axis about which V is symmetric, is conserved.