Physics Notes - Herong's Tutorial Notes

∟Introduction of Lagrangian

∟Lagrangian in Cartesian Coordinates

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

In previous sections, we have discussed Lagrangian with position, velocity, and acceleration as scalar values, which is valid only if the object is moving in a 1-dimensional straight line.

Now, let's re-define Lagrangian for a single object moving a 3-dimensional space using Cartesian coordinates.

First the position of an object as a function of time can be expressed as a vector function, r(t): 

r(t) = (x(t), y(t), z(t))

or:
  r = (x, y, z)                    
  r = (rx, ry, rz)

The velocity becomes a vector function of time, r'(t): 

r' = dr/dt
  = (drx/dt, dry/dt, drz/dt)

The acceleration also becomes a vector function of time, r"(t): 

r" = dr'/dt
  = (dr'x/dt, dr'y/dt, dr'z/dt)
  = (d2x/dt2, d2y/dt2, d2z/dt2)

Lagrangian Function in Cartesian Coordinates

Now let's look at the Lagrangian function in Cartesian coordinates. 

H = T - V                          (G.1)

The kinetic energy part can be expressed in terms of r' 

T = 0.5*m*|r'|2                   (G.21)

The potential energy part can be expressed in terms of r: 

V = V(r)                          (G.22)

So the Lagrangian function can be expressed in terms of r and r' directly, and t indirectly: 

L = T - V                          (G.1)

or:
  L = 0.5*m*|r'|2 - V(r)          (G.23)
    # G.21 and G.22 applied

In general, the Lagrangian function can be expressed in terms of r, r', and t: 

L = L(r,r',t)                     (G.24)
  # where r and r' are vectors of 3 coordinates.

If we apply the Hamiltonian definition to Lagrangian, we can have: 

L = T - V

or:
  L = 2*T - H
    # H = T + V applied

or:
  L = m*|r'|2 - H
    # T = 0.5*m*|r'|2 applied

or:
  L = p∙r' - H
    # p is the momentum defined as p = m*r'
    # ∙ is the vector dot product

The last equation is called Legendre transformation, which converts Hamiltonian to Lagrangian, and vice versa.

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

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