Examples etc Flashcards
What does is meant by homogeneity of space?
The motion is independent of position (i.e. no v(r))
How can we prove that space is homogeneous?
- 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
How do we express the Lagrangian after it has been displaced by 𝛿r?
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
What is ∇L equal to if 𝛿L = 0 for and 𝛿r?
∇L = 0, so dL/dqi = 0
What do we find from the E-L equation from dL/dqi = 0?
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
How can we express 𝛿L for many particles?
𝛿L = sum over a of dL/dra . 𝛿r, where a is the index of the particle
If 𝛿L = 0 for any 𝛿r, what do we get the sum of dL/dra is equal to?
sum over a of dL/dra = 0
What can we do with the E-L equation for many particles?
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
What do we find from the sum of the E-L equations?
p(a) = dL/dr’a, so total momentum P = sum over a of p(a) -> P is conserved if space is homogeneous
What is the one exception for the homogeneity of space with velocity depending on r?
Can hold if there are interactions between particles i.e. v(r2-r1)
What is the conservation law for the momenta?
Any canonical momenta pi whose conjugate coordinate qi does not appear in L is conserved.
What is meant by the homogeneity of time?
If L is not an explicit function of time, it doesn’t matter what time motion occurs.
What can we infer from the Homogeneity of time?
If time is homogeneous, L = L(qi, qi’), so dH/dt = 0, H is conserved (E is conserved)
What is meant by the isotropy of space?
Mechanical properties don’t change on rotating the system if space is isotropic. Conversely, special direction effects motion.
What do we do to show the isotropy of space?
- 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
What is the change in Lagrangian for a single particle when rotation the space?
𝛿L = sum over i of (dL/dqi * 𝛿qi + dL/dqi’ *𝛿qi’) = dL/dr 𝛿r + dL/dv 𝛿v
What is another way of writing dL/dqi and how can we write this?
Fi = ρi’ = dL/dqi, so 𝛿L = ρi’ . 𝛿r + ρi . 𝛿v
What do we then substitute into 𝛿L?
- 𝛿r = 𝛿Ф Xr and 𝛿v = 𝛿Ф X v
- Can then rearrrange using law to 𝛿L = 𝛿Ф.(rXρ’ + vXρ) = d/dt(rXρ)
What do we finally find from 𝛿L?
- 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
What is the change in Lagrangian for many particles when rotation the space?
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)
What is the next step after finding the in Lagrangian for multiple particles?
- 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
What is the final version of 𝛿L for many particles?
𝛿L = 𝛿Ф*(sum over a of d/dt(r(a)Xρ(a)) = 0
What do we find from the final version of 𝛿L for many particles?
sum over a of d/dt(r(a)Xρ(a)) = 0 so total angular momentum of system is conserved if space is isotropic
What can we summarise from these Lagrangian rearrangements?
With n generlised coordinates, have up to 2n+1 conserved quantities (n linear momenta, n angular momenta and energy)
What is Noether’s theorem?
If a symmetry in the Lagrangian exists, there is a corresponding constant of the motion.
What is a good example to show this theory?
-Field is an infinite homogeneous plane, e.g. V constant at height z i.i. V(z), but T(x’, y’, z’)
How can we find the conserved quantities in this example?
- 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)
How can we find the conserved quantities for a field due to 2 point objects?
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.
