Physics Notes - Herong's Tutorial Notes

∟Introduction of Lagrangian

∟Lagrange Equations in Cartesian Coordinates

This section provides a quick introduction to the Lagrange Equations in Cartesian coordinates, expressed in terms of r, r' and t.

If the Lagrangian function is expressed in Cartesian Coordinates as shown in the last section, the Action, Hamilton's Principle and Lagrange Equations can also be expressed in Cartesian Coordinates.

Action in Cartesian Coordinates

If the Lagrangian function is expressed in Cartesian Coordinates, the Action can also be expressed in Cartesian Coordinates. 

S = S[r(t)]

or:
  S[r(t)] = ∫ L(r,r',t)dt          (G.25)

Hamilton's Principle in Cartesian Coordinates

In Cartesian Coordinates, the Hamilton's Principle stay the same: the true position function r(t) of a system is a stationary point of the Action S[r(t)] functional.

Hamilton's Principle is also called Stationary Action.

Lagrange Equations in Cartesian Coordinates

Now the Lagrange Equation becomes the Lagrange Equations in Cartesian Coordinates: 

d(∂L/∂r')/dt = ∂L/∂r               (G.26)

or:
  d(∂L/∂r'x)/dt = ∂L/∂rx
  d(∂L/∂r'y)/dt = ∂L/∂ry
  d(∂L/∂r'z)/dt = ∂L/∂rz

We can prove that Lagrange Equations is equivalent to Hamilton's Principle in Cartesian Coordinates in the same way as we did earlier in the 1 dimension case.

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

 What Is Lagrangian

 Action - Integral of Lagrangian

 Action - Functional of Position Function x(t)

 Hamilton's Principle - Stationary Action

 What Is Lagrange Equation

 Other Proofs of the Lagrange Equation

 Lagrange Equation on Free Fall Motion

 Lagrange Equation on Simple Harmonic Motion

 Lagrangian in Cartesian Coordinates

►Lagrange Equations in Cartesian Coordinates

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait

 References

 Full Version in PDF/ePUB