Course problem bank

Problems across the course

These problems combine methods and assumptions from different chapters. The chapter exercise sets provide additional practice with each individual concept. Hints and worked solutions are on a separate page.

Geometry and work

1. F1 A bead of mass $m$ moves on the rising parabola $y=cx^2+Ut$ under gravity. Use $q=x$ to find the velocity, virtual displacement, kinetic energy, generalized gravity force, and equation of motion.

2. F2 A pendulum is described by arc coordinate $s=a\theta$. Construct $L$, its momentum, and $H$, and compare with angular coordinates.

Lagrangians and forces

3. F3 Two Atwood masses are joined over a pulley with moment of inertia $I$ and radius $R$, with no slip. Find the acceleration.

4. F4 A pendulum with small angular displacement has viscous torque $-b\dot\theta$ and driving torque $\tau_0\cos\Omega t$. Construct its equation and persistent amplitude.

5. F5 For $L=\tfrac12m(\dot x^2+\dot y^2)-\tfrac12k(x-y)^2$, identify a cyclic coordinate and solve the two independent motions.

Hamiltonians and state space

6. F6 Construct $H$ and its equations for $L=\tfrac12a(q)\dot q^2-V(q)$, and verify the Lagrange equation.

7. F7 A free particle is described by $x=q+X(t)$. Find the Hamiltonian and equations; explain its relation to laboratory momentum.

8. F8 Describe the pendulum phase portrait on the two sides of energy $2mga$, and explain the role of a string constraint.

Radial motion and stability

9. F9 For $V(r)=ar^n$ with $an>0$, find the circular condition and local frequency ratio.

10. F10 A potential has $V_{\rm eff}(r_c+\eta)=E_c+A\eta^4+O(\eta^5)$ with $A>0$. What can be said about local radial stability?

Kepler orbits and timing

11. F11 For $\mu=k=1$, a particle starts at $\mathbf r=(1,0)$ with velocity $(0,\sqrt{3/2})$. Find $E,\ell,e,a,r_a$, and period.

12. F12 Two equal masses $m$ move on a circular relative orbit of separation $d$. Find the period and each body’s speed about the center of mass.

13. F13 An ellipse has the same angular momentum as a circle of radius $r_c$ but eccentricity $e=0.8$. Find its semimajor axis and period ratio to that circle.

Runge–Lenz vector and encounters

14. F14 For $\mu=k=1$, $\mathbf r=(1,0,0)$ and $\mathbf p=(1/2,1,0)$, calculate the eccentricity vector and periapsis direction.

15. F15 How must impact parameter change to keep the same attractive deflection if incoming speed doubles?

16. F16 Explain which conserved quantities remain when a small time-independent radial perturbation is added to inverse-square attraction.

Chapter problem collections