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

2024
Views:

360

Pages:

5
Author:

Jabin F.

Generalized Coordinates Suitable variables chosen to describe the dynamics of a system are called generalized coordinates. For a particle moving in the xy-plane, (x, y) coordinates are generalized coordinates. For a simple pendulum, we find the use of θ to be convenient. So, θ is the generalized coordinate. For electric circuits, charge is a generalized coordinate. For thermodynamic systems, P, V, T, etc. are generalized coordinates. Example: Simple pendulum (length is constant, bob oscillates in a fixed vertical plane) Generalized coordinates: (x, y) There is a relation between x and y: l² = x² + y² (l is independent of time) So, only one variable is independent. Constraint is holonomic and scleronomic. Let us use polar coordinates: x = l sinθ y = l cosθ l is constant. ⇒ θ is the only variable. ⇒ θ is independent of time. Generalized coordinate = θ **Example 1: Particle flying inside a sphere of fixed radius R** x² + y² + z² = R² x, y, z are independent, d.o.f = 3. Constraint is Non-Holonomic & Scleronomic. **Example 2: Particle moving on a sphere surface** x² + y² + z² = R² As R is known, only 2 independent variables. d.o.f = 3, generalized co-ordinates (x, y, z) or (y, z) or (x, z) Constraint is holonomic & scleronomic. Let us use spherical polar co-ordinates (r, θ, φ) Here r = R constant; so θ and φ are only 2 variables. d.o.f = 2.