University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

291

Pages:

5
Author:

Jabin F.

Hamiltonian (H): Obtained by Legendre transformation of Lagrangian. Given as: H(qi, pi, t) = Σ(pi * qi - Li) Transformation from L to H involves a change of variable from qi to pi. Sometimes H is equal to energy, but not always. Partial Derivatives of H: Taking partial derivatives of H with respect to qi and pi: ∂H/∂qi = pi ∂H/∂pi = qi Problem P-97: Three particles lying on a smooth horizontal surface are connected to each other as shown in the figure. The problem asks which of the following options correctly represents the longitudinal normal mode of vibration corresponding to frequency ω: (a) 000 →←→ (b) 000 →←→ (c) 000 ←→→ (d) 000 ←←→ Matrix Equation: [k-w²m -k 0 -k] [ -k 2k-w²m -k] [ 0 -k k-w²m] Finding Relation Between Displacements: [k-w²m -k 0 -k] [x1 x2 x3] = 0 [ -k 2k-w²m -k] [ 0 -k k-w²m] Simplified Equations: (k-w²m)x1 - kx2 = 0 -kx1 + (2k-w²m)x2 - kx3 = 0 -kx2 + (k-w²m)x3 = 0 Solving Equations: x1 = x3 x2 = 0