Lagrangian Dynamics UPDATED ACTUAL
Exam Questions and CORRECT Answers
What is a holonomic constraint? - CORRECT ANSWER - An algebraic equation between
co-ordinates. Each holonomic constraint reduces the number of degrees of freedom by 1. Explicit
t dependence is allowed.
What are Lagrange's equations in their most general form? - CORRECT ANSWER - d/dt
(∂T/∂ẋj) - ∂T/∂xj = Qj
What are Lagrange's equations for conservative forces? - CORRECT ANSWER - d/dt
(∂L/∂ẋj) - ∂L/∂xj = 0
For a rolling object, what is the relation between its linear displacement and its angular
displacement? - CORRECT ANSWER - x = θr
For pulleys, what is a useful way to find holonomic constraints? - CORRECT ANSWER -
Use the fact that the lengths are fixed and the height of some stuff is fixed.
For object 1 moving on or in object 2, what is the speed of object 1? - CORRECT
ANSWER - v1 = v1in2 + v2
What is the length of a curve? - CORRECT ANSWER - ∫ √[1+(dy/dx)^2] dx
What is the Euler equation and how is it derived? - CORRECT ANSWER - d/dx (∂F/∂y') -
∂F/∂y = 0 , Consider a small change in path η, same fixed endpoints as before. Taylor expand,
define δI = I[y+η]-I[y]. Integrate second term by parts, set δI equal to zero.
What are the first integrals of Euler's equation? - CORRECT ANSWER - F not of y: ∂F/∂y'
= C , F not of x: y'∂F/∂y' - F = C (Proof by total derivative of F wrt x, Euler's eqn, rewrite reverse
product rule.)
Exam Questions and CORRECT Answers
What is a holonomic constraint? - CORRECT ANSWER - An algebraic equation between
co-ordinates. Each holonomic constraint reduces the number of degrees of freedom by 1. Explicit
t dependence is allowed.
What are Lagrange's equations in their most general form? - CORRECT ANSWER - d/dt
(∂T/∂ẋj) - ∂T/∂xj = Qj
What are Lagrange's equations for conservative forces? - CORRECT ANSWER - d/dt
(∂L/∂ẋj) - ∂L/∂xj = 0
For a rolling object, what is the relation between its linear displacement and its angular
displacement? - CORRECT ANSWER - x = θr
For pulleys, what is a useful way to find holonomic constraints? - CORRECT ANSWER -
Use the fact that the lengths are fixed and the height of some stuff is fixed.
For object 1 moving on or in object 2, what is the speed of object 1? - CORRECT
ANSWER - v1 = v1in2 + v2
What is the length of a curve? - CORRECT ANSWER - ∫ √[1+(dy/dx)^2] dx
What is the Euler equation and how is it derived? - CORRECT ANSWER - d/dx (∂F/∂y') -
∂F/∂y = 0 , Consider a small change in path η, same fixed endpoints as before. Taylor expand,
define δI = I[y+η]-I[y]. Integrate second term by parts, set δI equal to zero.
What are the first integrals of Euler's equation? - CORRECT ANSWER - F not of y: ∂F/∂y'
= C , F not of x: y'∂F/∂y' - F = C (Proof by total derivative of F wrt x, Euler's eqn, rewrite reverse
product rule.)
