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

2024
Views:

154

Pages:

6
Author:

Jabin F.

Generalized Force (Qi): It could represent force, torque, or other quantities. The generalized force Qi is defined such that its product with the change in the generalized coordinate (dqi) gives the work done. Work done: dw = Qi * dqi Relation to Actual Force and Displacement: In terms of actual force (Fi) and actual displacement (dli), work done is written as: dw = Fi * dli Comparing the two expressions for work done, we get: Qi * dqi = Fi * dli Therefore, the relation between generalized force and actual force is: Qi = Fi * dli / dqi Examples of Coordinate Systems: Cartesian System (x, y, z; h1 = 1, h2 = 1, h3 = 1): Q1 = Fx Q2 = Fy Q3 = Fz Spherical Polar System (r, θ, φ; h1 = 1, h2 = r, h3 = r sinθ): Q1 = Fr Q2 = rFθ Q3 = r sinθ Fφ The Hamiltonian of a 1D system is given as: H = (1/2)mẋ² + (1/2)kx² where m and k are constants. Solution Methods: Method 1: Using Hamilton's Equations of Motion (EOM) Hamilton's equations are used to derive the equations of motion: ẋ = ∂H/∂p, ṗ = -∂H/∂x After substituting the Hamiltonian and performing the derivatives, the following equation is obtained: mẍ + kx = 0 Method 2: Using Lagrange's EOM The Lagrangian L is derived from the Hamiltonian: L = T - V = (1/2)mẋ² - (1/2)kx² Lagrange's equations of motion are applied: d/dt(∂L/∂ẋ) - ∂L/∂x = 0 After performing the derivatives and simplifications, the same equation of motion is obtained: mẍ + kx = 0 Problem P-102: Given: p = 1/2(r + θ) q = 1/2(r - θ) Task: Find F(p, q). Solution: Step 1: Express r and θ in terms of p and q: r = p + q θ = p - q Step 2: Calculate the Jacobian determinant: ∂(r,θ)/∂(p,q) = |(∂r/∂p ∂r/∂q), (∂θ/∂p ∂θ/∂q)| = |(1 1), (1 -1)| = -2 Step 3: Use the Jacobian determinant to find F(p, q): F(p, q) = F(r(p, q), θ(p, q)) / |∂(r,θ)/∂(p,q)| = F(p+q, p-q) / 2 Problem P-103: Task: Which of the following geometric function F(p, q) represents rotation in phase space through an angle θ? Solution: Option 1: p' = p cosθ + q sinθ q' = -p sinθ + q cosθ This transformation represents a rotation in phase space.