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Paper I › Classical Mechanics ›Lagrangian & Hamiltonian Mechanics
Lagrangian & Hamiltonian Mechanics
3 Questions
2 Solutions Available
1 Coming Soon
67% solutions available
2025
UPSC
10 MarksHardPaper I
Q 72. Normalised wave function of hydrogen for 1s state is $\psi_{100}=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}$
, where $a_0=\frac{\hbar^2}{me^2}$ being the Bohr radius. Calculate the expectation value of potential energy in this state.
2024
UPSC
20 MarksMediumPaper I
Derive the Euler-Lagrange quations of motion
This is a standard derivation frequently asked in UPSC Physics Optional. The key is to set up the formalism correctly before computing.
Lagrangian: L = T − V | Euler-Lagrange: d/dt(∂L/∂q̇) − ∂L/∂q = 0
Step-by-step solution:
Define the system and identify generalised coordinates q₁, q₂, …
Write expressions for kinetic energy T and potential energy V.
Form the Lagrangian L = T − V.
Apply the Euler-Lagrange equation to each coordinate.
Simplify to obtain the equations of motion. Identify conserved quantities from cyclic coordinates.
📐 Diagram: Phase portrait / free body diagram would be drawn here in the actual answer sheet. Always include a neat diagram in UPSC answers.
UPSC Tip: For a 15-mark question, spend the first 2 minutes planning your answer structure: definition → derivation → result → significance/application.
