From energy to motion

Science · Forces & Motion · ages 21-22

Name ______________________ Date ____________

With no nonconservative generalized force, a coordinate never appears in L. What does that imply?
- Its momentum is conserved
- Its acceleration must be zero
- The Lagrangian is invalid
What does applying the Euler Lagrange equation to L return?
- The constraint forces on the system
- Each coordinate's equation of motion
- The numerical value of the total energy
Which symmetry conserves angular momentum?
- Shifting the clock
- Rotation symmetry
- Rescaling the mass
Free motion in a plane uses polar coordinates with T = ½m(ṙ² + r²φ̇²) and V = 0. Which momentum is conserved for every free trajectory?
- The radial momentum m r dot is conserved
- Neither momentum is conserved
- The angular momentum m r squared phi dot
For the pendulum Lagrangian script L equals half m l squared rate squared plus m g l cos theta, what is partial L partial rate?
- m l squared rate
- m g l sin theta
- half m l squared rate squared
With no nonconservative generalised force in a cyclic coordinate, its conjugate momentum is conserved.
Circle one: True False
An autonomous Lagrangian has no explicit time dependence and no nonconservative generalized forces. What follows?
- The sum over all coordinates of q dot times partial L partial q dot, minus script L, is conserved
- T + V is always the conserved quantity in every coordinate system, whatever the rates or coordinates are
- Only cyclic coordinates yield any conserved quantity
From script L equals half m l squared rate squared plus m g l cos theta, the Euler Lagrange equation in theta gives which equation?
- theta double dot plus g over l times theta equals zero for all swings
- theta dot squared plus g over l cos theta equals zero
- theta double dot plus g over l sin theta equals zero

Answer key

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From energy to motion W1-mt_GSEFABbEXI-s1

Its momentum is conserved · A missing coordinate is cyclic, so with no nonconservative generalized force its conjugate momentum stays constant.
Each coordinate's equation of motion · Each coordinate yields its own equation of motion through the fixed recipe.
Rotation symmetry · Rotating the description changes nothing, so the corresponding momentum coasts.
The angular momentum m r squared phi dot · Only phi is absent from the whole Lagrangian, so the angular momentum m r squared phi dot is conserved; r appears in the kinetic energy, so the radial momentum need not be conserved, since a straight radial trajectory keeps m r dot constant.
m l squared rate · Only the kinetic term carries the rate, so partial L partial rate is m l squared rate; the cosine term has no rate and drops out.
True · Cyclic means the coordinate never appears in L, so the Euler Lagrange equation leaves its conjugate momentum fixed when no nonconservative generalised force acts.
The sum over all coordinates of q dot times partial L partial q dot, minus script L, is conserved · With no explicit time dependence and no nonconservative generalized forces, the unforced Euler-Lagrange equation makes the generalized energy E = Σᵢ q̇ᵢ ∂L/∂q̇ᵢ − L constant. It equals T + V when L = T − V with T homogeneous quadratic in the rates and V rate-independent, so the two are not always the same thing.
theta double dot plus g over l sin theta equals zero · The derivatives give ml²θ̈ + mgl sin θ = 0, which simplifies to θ̈ + (g/l) sin θ = 0; the small-angle form θ̈ + (g/l)θ = 0 holds only for small swings.