Variational Principles

Question 1

What are 2 things which variational principles give us?

Answer 1

• Let us extremise quantities such as path lengths or functions
• Give us global view of path a system adapts e.g. to minimise time taken to get from A to B

Question 2

What is the lifeguard example of a variational principle?

Answer 2

Want to rescue swimmer so need to take quickest path, which is not necessarily a straight line to swimmer, and you are slower in the water.

Question 3

What is the optics example of a variational principle?

Answer 3

Light does not travel in a straight line, it extremises the optical path.

Question 4

What is Fermat’s principle?

Answer 4

total optical path length L = integral of n(r) ds, where n is the refractive index, r is the position and s is the unit of length.

Question 5

What are the steps to derive Snells law using variational principals?

Answer 5

• Path length of light = n1d1+n2d2, where d is the length of the path
• Use pythagoras to split d into h and x
• Look for minima in L as we vary x by finding dL/dx and set = 0
• Use trig to find sin terms
• Snells law

Question 6

How do we find the actual path something takes when it moves from A to B?

Answer 6

A = integral of L dt, where L is the Lagrangian

Question 7

State Hamilton’s principle.

Answer 7

Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval, the actual path followed is that which minimises the time integral of the difference between the kinetic and potential energies.

Question 8

What is the classical mechanics example of the Lagrangian?

Answer 8

L = T-V

Question 9

What is the relativistic motion example of the Lagrangian?

Answer 9

L = -m0c^2/γ = -m0c^2*sqrt(1-(v^2/c^2))

Question 10

What is a good example of minimising the path A?

Answer 10

Mass m, moving in 1D, under influence of gravity. Launched upwards at time t=0 from x=0.

Question 11

What are the first two steps in minimising the path A for the mass under gravity problem?

Answer 11

• Find the Lagrangian (T-V = 1/2mv^2 - mgx)

- Define start and end position: x(t=0) = 0, x(t=τ) = X

Question 12

What are the 3rd and 4th steps in minimising the path A for the mass under gravity problem?

Answer 12

• Construct path that satisfies boundary conditions and is physical: e.g. x(t) = X/τ * t, so try x(t) = X/τ * t + b(τ-t)t
• Use x(t) to get Lagrangian (by differentiating x(t) to get velocity)

Question 13

What are the 5th and 6th steps in minimising the path A for the mass under gravity problem?

Answer 13

• Use L to get A by integrating from 0 to τ
• Minimise A by finding the constant b which minimises the action (set dA/db = 0 and rearrange for b), then sub b back in