What Is Generalized Momentum

This section provides a quick introduction to generalized momentum, which is defined as a vector of partial derivatives of Lagrangian against generalized velocity components.

If we want to express Hamiltonian in terms of position and momentum in generalized coordinates, we needed to transform the ordinary momentum of p = m*v into the generalized momentum.

One way to define the generalized momentum is to use partial derivatives of Lagrangian against generalized velocity.

p = ∂L/∂q'
  # p is the generalized momentum
  # q' is the generalized velocity
  # L is the Lagrangian

As a comparison, here is the definition of the ordinary momentum.

p = m*r'
  # p is the mechanical momentum
  # m is the mass 
  # r' is the velocity in Cartesian coordinates

Note that we are using the sample symbol p to represent both mechanical momentum and generalized momentum. You need to differentiate them by looking the context where it is used.

By the way, the ordinary momentum is also called the mechanical momentum or kinematic momentum.

If we use Cartesian coordinates for conservative system, the generalized momentum is the same as the ordinary momentum as shown below.

p = ∂L/∂q'

  p = ∂L/∂r'
    # q' = r' in Cartesian coordinates

  p = ∂(0.5*m*|r'|2 - V(r))/∂r'
    # Since L = 0.5*m*|r'|2 - V(r) 

  p = ∂(0.5*m*|r'|2)/∂r'
    # Since ∂(V(r))/∂r' = 0 

  p = m*r'
    # It matches the ordinary momentum definition.

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

 Introduction of Lagrangian

Introduction of Generalized Coordinates

 Generalized Coordinates and Generalized Velocity

 Simple Pendulum Motion in Generalized Coordinates

 Hamilton's Principle in Generalized Coordinates

 Lagrange Equations in Generalized Coordinates

 Lagrange Equations on Simple Pendulum

What Is Generalized Momentum

 What Is Legendre Transformation

 Hamilton Equations in Generalized Coordinates

 Phase Space and Phase Portrait


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