Setup: A bead of mass (m) slides without friction on a circular hoop of radius (R). The hoop rotates with constant angular velocity (\omega) about a vertical axis. Let (\theta) be the angle from the vertical (top of hoop).
(a) Find Lagrangian.
(b) Determine equilibrium angles.
Looking for a clear, structured PDF of problems and worked solutions in Lagrangian mechanics? Here's a concise guide and resources you can use to create or find one.
Setup: Mass ( m ) attached to a massless rod of length ( l ), swinging under gravity.
Generalized coordinate: ( \theta ) (angle from vertical)
Kinetic energy: ( T = \frac12 m (l\dot\theta)^2 )
Potential energy: ( U = -mgl \cos\theta ) (zero at bottom)
Lagrangian: ( L = \frac12 m l^2 \dot\theta^2 + mgl \cos\theta ) lagrangian mechanics problems and solutions pdf
Euler-Lagrange:
[
\fracddt(m l^2 \dot\theta) + mgl \sin\theta = 0 \quad \Rightarrow \quad \ddot\theta + \fracgl\sin\theta = 0
]
Solution hint: For small angles, ( \sin\theta \approx \theta ), giving simple harmonic motion.
Below are foundational problems found in most PDF resources. Each includes a short solution outline.
Chapter 1: Calculus of Variations
1.1 Shortest path between two points
1.2 Brachistochrone problem
1.3 Geodesic on a sphere Problem 1: Simple Pendulum Setup: A mass (m)
Chapter 2: Lagrange’s Equations – Basic Applications
2.1 Simple pendulum
2.2 Atwood machine
2.3 Bead on a frictionless wire (parabolic)
2.4 Block sliding on a moving wedge
Chapter 3: Central Forces & Constrained Motion
3.1 Particle in a central potential ( V(r) = -k/r )
3.2 Double pendulum (small oscillations)
3.3 Particle on a sphere (pendulum with variable length)
Chapter 4: Non‑holonomic Systems & Lagrange Multipliers
4.1 Disk rolling down an inclined plane
4.2 Hoop rolling without slipping on a horizontal rod
4.3 Particle constrained to a moving surface
Chapter 5: Noether’s Theorem & Symmetries
5.1 Translational symmetry → linear momentum
5.2 Rotational symmetry → angular momentum
5.3 Time translation symmetry → energy conservation ( \sin\theta \approx \theta )
Chapter 6: Small Oscillations & Normal Modes
6.1 Two coupled pendulums
6.2 Triple spring‑mass system
6.3 Molecular vibrations (linear triatomic molecule)
Appendices
A. Quick reference: Lagrangian mechanics formulas
B. Answers to selected problems (odd numbers)
C. Bibliography
Setup: A mass (m) attached to a massless rod of length (L). The rod pivots without friction. Use the angle (\theta) from the vertical.
(a) Find the Lagrangian.
(b) Derive the equation of motion.
(c) For small oscillations, find the period.
A high-quality PDF on this topic is typically used by upper-undergraduate or introductory graduate physics students (Classical Mechanics, PHYS 301–400 level). It should bridge the gap between theory (Lagrange’s equation: ( \fracddt \left( \frac\partial L\partial \dotq_j \right) - \frac\partial L\partial q_j = 0 )) and real problem-solving.