6. Conservation Laws

Question 1

Describe what the Lagrangian contains

Answer 1

All information on a systems dynamics

Question 2

What is another way of saying a quantity is conserved, and give an example?

Answer 2

It is an “integral of motion”

- e.g. H = E

Question 3

What are conservation laws derived on?

Answer 3

The uniformity of space and time

Question 4

What does homogeneous space mean?

Answer 4

The motion is independent of position

Question 5

What does homogeneous space mean in terms of the potential?

Answer 5

V(r) is not allowed

- Can have V(r_2 - r_1) (relative position)

Question 6

Does a small fixed displacement alter the motion of a system?

Answer 6

No

Question 7

See p5 of document for the diagram of homogeneous space

Answer 7

Do it

Question 8

See page 5 of document for the single particle example

Answer 8

Do it

Question 9

See 6.1.2 for the many particle example

Answer 9

If u want lol

Question 10

Apply newton’s first law to free space

Answer 10

Linear momentum is conserved in free space

Question 11

Is momentum additive?

Answer 11

Yes

Question 12

Apply Newton’s third law to homogenous space

Answer 12

The sum of all forces is equal to 0

- see p6 of document

Question 13

State the relationsihp between p_i, q_i and the Lagrangian

Answer 13

Any canonical momenta p_i whose conjugate coord q_i doesn’t appear in its Lagrangian is conserved

Question 14

What can be said about the Lagrangian and being a function of time?

Answer 14

If it isn’t an explicit function of time, the time motion is irrelevant

Question 15

Define isotropy

Answer 15

Uniformity in all directions

Question 16

If a system is isotropic, do its mechanical properties change?

Answer 16

No

Question 17

See 6.3, 6.3.1 and 6.3.1 for isotropy of space and how angular momentum is conserved for single and many particles

Answer 17

Do it

Question 18

State Noether’s theorem

Answer 18

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

Question 19

Up to how many conserved quantities can we have for n generalised coordinates?

Answer 19

2n + 1

• n linear momenta
• n angular momenta
• energy