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

2024
Views:

472

Pages:

5
Author:

Jabin F.

Problems on types of dynamics of System for given Lagrangian H: L is not uniquely defined. Types of dynamics cannot be concluded by looking at L or H. To know dynamics of the system, write the equation of motion and obtain q_i (acceleration). Following conclusions will arise: q_i Dynamics q_i = 0 Free particle q_i = -ω^2q_i Simple harmonic oscillator q_i = -f(q_i) Oscillatory q_i = f(q_i) Non-oscillatory q_i = -f(q_i) + f(t) Non-oscillatory (both even or odd) q_i = -cq_i - γq̇ + f(t) Damped oscillations (damping term) q_i = -cq_i - γq̇ + f(t) Forced oscillation (f(t) is independent of q, q̇) Problem P-13: The dynamics of a particle governed by the Lagrangian L = 1/9 m(dot x)^9 - 1/2 kx^9 - kx(dot x)t describes: Solution: d/dt(∂L/∂(dot x)) - ∂L/∂x = 0 d/dt(mx^8 - kx^8t) - (-kx^8 - kx^7t) = 0 mx^7 - kx^7t - kx^8 + kx^8t + kx^7t = 0 mx^7 = 0 x = 0 (free particle) Problem P-13: The Lagrangian of a particle L = q̇^8 - q(dot q) describes the dynamics of: Solution: d/dt(∂L/∂(dot q)) - ∂L/∂q = 0 d/dt(8q̇^7 - q̇) - (-q̇) = 0 8q̇^6 - q̇ + q̇ = 0 q̇ = 0 (free particle) Problem D-17-7/8/19: Given L = e^(γt)(1/8 m(dot x)^8 - 1/2 kx^8), what is the type of dynamics of the system? Solution: d/dt(∂L/∂(dot x)) - ∂L/∂x = 0 d/dt(me^(γt)(dot x)^7 - kxe^(γt)) - (-kxe^(γt)) = 0 m(dot x)^7e^(γt) + γme^(γt)(dot x)^7 + kxe^(γt) + kxe^(γt) = 0 (dot x)^7 + γ(dot x)^7 + (k/m)x = 0 (dot x)'' + γ(dot x)' + (k/m)x = 0 Dynamics of the system: damped oscillation Problem P-13: The Lagrangian of a particle of mass m moving in one dimension is given by: L = 1/2 m(dot x)^2 - bx, where b is a positive constant. The coordinate of the particle x(t) at time t is given by: Solution: L = 1/2 m(dot x)^2 - bx EOM: d/dt(∂L/∂(dot x)) - ∂L/∂x = 0 d/dt(mx) - (-b) = 0 mx + b = 0 ẍ = -b/m Taking integration, we get: x = -bt/m + c1 Again taking integration, we get: x = -bt^2/(2m) + ct + c2