Extended Phase Space for Time-Dependent Hamiltonians

1. The Problem

In a time-dependent Hamiltonian system,

the equations of motion are

For an observable $f(q,p,t)$, the time evolution is

where the Poisson bracket only involves the phase space variables $(q,p)$. Because of the explicit $t$-dependence, the system is non-autonomous.

2. Making the System Autonomous

Introduce an extended phase space with a new canonical pair:

Define the extended Hamiltonian:

The Poisson bracket on the extended space is:

3. Hamilton’s Equations in Extended Space

Using $K=H+E$:

Let $\tau$ be the Hamiltonian evolution parameter in the extended space. Since $\frac{dt}{d\tau }=1$, we get $\tau =t+\text{const}$. Therefore, the Hamiltonian flow parameter coincides with physical time.

4. Time Evolution of Observables

For any function $F(q,p,t,E)$:

Expanding this:

5. Physical Observables

A physical observable from the original system does not depend on $E$:

Then:

which reproduces the original evolution law.

6. The Source of Confusion

In the extended phase space:

So $E$ generates translations in the coordinate $t$. However:
Key Distinction.
The derivative with respect to the coordinate $t$ is not the same as time evolution of the system. These are different operations.

7. Geometric Meaning of the Brackets

The key identity for physical observables:

8. Flow Interpretation

Observe that

But $X_E=\frac{\partial }{\partial t}$ only shifts the time coordinate without changing $(q,p)$. Thus:

• $X_H$ moves the system in phase space.

• $X_E$ advances the clock.

• $X_K$ does both — this is the true physical evolution.