GATE PH: Classical Mechanics

Classical Mechanics in GATE PH tests Lagrangian and Hamiltonian formulations, central-force motion, small oscillations, rigid-body dynamics, canonical transformations, and Poisson brackets. The exam favours problems where a clever generalised coordinate removes constraints and exposes a simple conserved quantity.

Core rules and concepts

• The Lagrangian L = T - V leads to the Euler-Lagrange equations d/dt(∂L/∂qdot) - ∂L/∂q = 0. Cyclic coordinates give conserved conjugate momenta.
• The Hamiltonian H = Σ p qdot - L is conserved when L has no explicit time dependence. Hamilton's equations are qdot = ∂H/∂p and pdot = -∂H/∂q.
• Central-force motion reduces to a one-dimensional effective potential V_eff = V(r) + L²/(2mr²). Circular orbits satisfy dV_eff/dr = 0.
• Small oscillations about equilibrium use the matrix equation det(V'' - ω²T'') = 0, where T'' and V'' are evaluated at equilibrium.
• Poisson brackets satisfy {q_i,p_j}=δ_ij; canonical transformations preserve them.

Worked example

For a simple pendulum of length l and mass m, L = ½ml²θdot² + mgl cosθ. For small θ, cosθ ≈ 1 - θ²/2, so L ≈ ½ml²θdot² - ½mglθ² up to a constant. Euler-Lagrange gives ml²θddot + mglθ = 0, hence θddot + (g/l)θ = 0. The angular frequency is ω = √(g/l).

Traps that cost marks

• Sign errors in the potential energy, especially for gravitational and spring systems.
• Treating constraint forces as generalised forces when they do no virtual work.
• Forgetting that normal-mode frequencies come from T'' and V'' matrices, not from L alone.
• Using non-inertial frames without adding fictitious forces.