Hamiltonian in Cartesian Coordinates

This section provides a quick introduction on Hamiltonian for a single object in Cartesian coordinates.

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

Now, let's re-define Hamiltonian 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)                    (H.7)
  r = (rx, ry, rz)                  (H.8)

The velocity becomes a vector function of time, v(t):

v = dr/dt
  = (drx/dt, dry/dt, drz/dt)

or:
  v = r'                           (H.9)
    # r' is a shorthand notation of first order derivative

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

a = dv/dt
  = (dvx/dt, dvy/dt, dvz/dt)
  = (d2x/dt2, d2y/dt2, d2z/dt2)

or:
  a = v'
  a = r"
     # r" is a shorthand notation of second order derivative

The momentum also becomes a vector function of time, p(t):

p = m*v
p = m*r'

Hamiltonian as a Function of (r,r')

Now let's look at the Hamiltonian function in Cartesian coordinates. And express it in terms of r and r':

H = T + V                          (H.1)

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

T = 0.5*m*|r'|2                   (H.10)

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

V = V(r)                          (H.11)

So the Hamiltonian can be expressed in terms of r and r' as:

H = T + V                         (H.1)

or:
  H = 0.5*m*|r'|2 + V(r)          (H.12)
    # H.10 and H.11 applied

Hamiltonian as a Function of (r,p)

We can also transform Hamiltonian to a function of position r and momentum p:

H = 0.5*m*|r'|2 + V(r)            (H.12)

or:
  H = 0.5*|p|2/m + V(r)           (H.13)
    # since p = m*r'

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

 What Is Hamiltonian

 Hamiltonian on Free Fall Motion

 Hamiltonian on Simple Harmonic Motion

 Hamiltonian on Simple Pendulum Motion

 What Is Momentum

 Relation of Momentum and Hamiltonian

Hamiltonian in Cartesian Coordinates

 Relation of Momentum and Potential Energy

 Hamilton Equations in Cartesian Coordinates

 Introduction of Lagrangian

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait

 References

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