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

2024
Views:

349

Pages:

5
Author:

Jabin F.

#Liouville's Theorem (2) If a system has degrees of freedom = f, then max no. of possible independent conserved quantities for system is 'f'. Explanation: Every EOM written in terms of coordinate is a 2nd order differential equation. Solution of this equation introduces two constants. So if dof is f, then there are f EOM. So solution of all EOM will introduce 2f constants. P-91 There exists a constant electric field in space in x-direction. A charged particle is thrown into the field in some arbitrary direction. Which of the following is conserved for the particle? (a) total linear momentum (b) Px (c) total angular momentum (d) Lz and energy Solution [L = \frac{1}{2}m(ẋ² + ẏ² + ż²) - qEz] [p_{x}, p_{y}, p_{z} \text{ is conserved due to } x, y, z \text{ are cyclic co-ordinates.}] [p_{x} \text{ is not conserved as } x \text{ is not cyclic.}] ∴ total linear momentum is not conserved. [L_{z} \text{ is conserved as } L \text{ is invariant under rotation about x-axis.}] [L_{x}, L_{y} \text{ are not conserved as } L \text{ varies w.r.t. x, y}] ∴ total angular momentum is not conserved. t is not explicitly present in Lagrangian, so energy is conserved. P-92 Lagrangian of a particle is L = (1/2)m(ẋ² + ẏ² + ż²) - q(a · r), where a is a constant vector. Which of the following is conserved? (a) only p_x (b) only p_y and p_z (c) only p_y (d) none of p_x, p_y, and p_z