Hamilton Equations in Cartesian Coordinates

This section provides a quick introduction to the Hamilton Equations in Cartesian Coordinates, partial derivative equations on the Hamiltonian function against position r and momentum p for an isolated conservative system.

What Are Hamilton Equations? Hamilton Equations are partial derivative equations on the Hamiltonian function against position r and momentum p for an isolated conservative system.

First let's take the partial derivative against position r.

```∂H/∂r = ∂T/∂r + ∂V/∂r
# This is a vector equation with 3 components

or:
∂H/∂r = ∂(0.5*|p|2/m)/∂r + ∂V(r)/∂r
# H.12 applied

or:
∂H/∂p = 0 + ∂V(r)/∂r
# Since 0.5*|p|2/m is independent of r
or:
∂H/∂r = -p'                     (H.15)
# H.14 applied
```

Now take the partial derivative against momentum p. It's very easy to do.

```∂H/∂p = ∂T/∂p + ∂V/∂p
# This is a vector equation with 3 components

or:
∂H/∂p = ∂(0.5*|p|2/m)/∂p + ∂V(r)/∂p
# H.12 applied

or:
∂H/∂p = ∂(0.5*|p|2/m)/∂p + 0
# Since V(r) is independent of p

or:
∂H/∂p = p/m
# The chain rule for derivatives applied

or:
∂H/∂p = r'                      (H.16)
# H.7 applied
```

Putting them together, we have the Hamilton Equations for an isolated conservative system:

```∂H/∂r = -p'                       (H.15)
∂H/∂p = r'                        (H.16)
```

Again Hamilton Equations are also called Hamilton's Law of Motion, which is equivalent to Newton's Second Law of Motion, which is equivalent to the Law of Conservation of Energy.

Now we can describe a moving object without using force and acceleration:

• Speed is the change of energy resulted from the change of momentum.
• The change of momentum is negative to the change of energy resulted from the change of position.

By the way, many text books derive Hamilton Equations from Lagrangian Equations. They are actually equivalent, as we will see later.