Lagrangian Methods in 1D

Question 1

What can we use calculus of variations for?

Answer 1

Instead of guessing the path, we can find function which extremises an integral.

Question 2

What is the equation for the action when trying to find the minimal path?

Answer 2

A = integral from t0 to t1 of L(x(t), v(t), t) dt

Question 3

How can we show the path varying in the variables?

Answer 3

x(t) -> x(t) + a(t) and hence v(t) -> v(t) + a’(t)

Question 4

What do we do to the end points of the path and what does this mean for the Lagrangian?

Answer 4

We fix the end points: a(t0) = a(t1) = 0, so L(x, v, t) -> L(x+a, v+a’, t)

Question 5

Using Taylor expansion, what does the first order of the Lagrangian for very small a(t) become?

Answer 5

L(x,v,t)+a(t)dL/dx + a’(t)dL/dv

Question 6

What does the equation for the action become?

Answer 6

The new Lagrangian term subbed in so: integral from t0 to t1 of L(x,v,t)+a(t)dL/dx + a’(t)dL/dv dt

Question 7

What is 𝛿A, the added on part?

Answer 7

New part without old part of the Action equation: integral from t0 to t1 of a(t)dL/dx + a’(t)dL/dv dt

Question 8

What can we do to simplify 𝛿A?

Answer 8

Integrate the second term (involving a’(t)) by parts, and then use a(t0) = a(t1) = 0, so this integrated part is zero.

Question 9

What is 𝛿A after the integration by parts? What do we do with this?

Answer 9

𝛿A = integral from t0 to t1 of a(t)*(dL/dx - d/dt(dL/dx’))

With this, for A to be stationary, we need 𝛿A = 0, so part inside the brackets = 0.

Question 10

What is the Euler-Lagrange equation?

Answer 10

dL/dx - d/dt * dL/dx’ = 0

Question 11

How can we use the E-L equation in the mass under gravity problem?

Answer 11

Find the different terms in the E-L equation using differentiation: find that it is equal to -mg-ma = 0, so a = -g, as expected from Newtons second law.

Question 12

What do we start with for the alternative derivation of the E-L equation?

Answer 12

An alternative path which differs from the optimum path: x(α,t) = x(0,t)+α*Ν(t), where Ν(t) is an arbitrary function

Question 13

What do we do after starting the alternative derivation of the E-L equation?

Answer 13

• Find the new Lagrangian is: L(x+αΝ, v+αΝ’, t)

- Know that for all Ν(t), path minimised for α = 0 as x(0,t) is optimum path (dA/dα at α = 0 = 0)

Question 14

What is the integral from t0 to t1 of L(t,x,v) dt equal to?

Answer 14

= f(x,v)

Question 15

How do we find dA/dα for the alternative derivation of the E-L equation?

Answer 15

• dA/dα = d/dα*integral from t0 to t1 of L dt = df/dα

- Use chain rule so df/dα = df/dx * dx/dα + df/dv * dv/dα

Question 16

What do we do with df/dα after using the chain rule?

Answer 16

• Sub in the Lagrangian integral again and rearrange so integral is on outside of brackets
• Use equations for x and v to differentiate
• Integrate by parts

Question 17

What do we get after integrating by parts?

Answer 17

• dA/dα = integral from t0 to t1 of Ν*(dL/dx - d/dt * dL/dv)
• if dA/dα at α = 0 = 0 for any Ν(t), we must have: dL/dx - d/dt * dL/dv = 0
• The E-L equation

Question 18

How do we use the E-L equation in a 1D potential, potential energy V(x)? (find Newtons second law)

Answer 18

• L = T-V = 1/2 * m*v^2 - V(x)
• Apply E-L: dL/dx = -dV/dx = force due to potential, d/dt(dL/dv) = dρ/dt
• F = dρ/dt = Newtons second Law

Question 19

For a swinging pendulum, what do we make the Lagrangian a function of and why?

Answer 19

L(θ, θ’), since the length is fixed, we can describe the position using only θ, don’t need x,y.

Question 20

What is the potential energy V equal to for the pendulum?

Answer 20

V = const - mgLcosθ

Question 21

What is the kinetic energy T equal to for the pendulum?

Answer 21

1/2 * m*v^2 = 1/2 * m(Lθ’)^2, where v = (Lθ’)^2

Question 22

What do we do once we have found the Lagrangian for the simple pendulum?

Answer 22

Sub it into the E-L equation and find the differentials, then rearrange to find an equation for simple harmonic motion: θ’’ = -g/L * θ

Question 23

For a rollercoaster, what can we use instead of x,y and why?

Answer 23

Can use arc length s(t), since it is a fixed path and x&y aren’t independent.

Question 24

What is the Lagrangian equal to for the roller coaster?

Answer 24

L = 1/2 * m*s’^2 - mgh(s) = L(s,s’)

Question 25

What do we do with the Lagrangian for the roller coaster problem?

Answer 25

Sub it into the E-L equation, and find s’’ = -g * dh/dm which is the equation of motion, so if we know h(s), we get s(t).

Question 26

What is the generalised version of the E-L equation?

Answer 26

dL/dq 0 d/dt * dL/dq’ = 0, where q(t) is a generalised coordinate and may be x(t), θ(t) etc, and q’ is the related velocity.