Introduction to Hamiltonian systems@

What is a Hamiltonian anyway?@

This is a question that comes up frequently when discussing concepts and ideas of dynamical systems theory for the first few times.

  • At a simple level, a Hamiltonian is a scalar function that generates dynamics taking the particular form of Hamilton’s equations:

\[\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q}.\]

Systems that follow this description are called Hamiltonian. In other words, the differential equations governing Hamiltonian systems are fully known if the system’s Hamiltonian \(H\) is given.

Example

Consider the Hamiltonian \(H = -1/2p^2\). The dynamics of this system are

\[\dot{q} = \frac{1}{p^3}, \quad \dot{p} = 0.\]

It’s not at all obvious, but the keen-eyed reader may spot that if \(p = \sqrt{a/GM}\), then \(H\) is actually the total energy of the two-body problem, and also that \(q\) is the mean anomaly.

Canonical variables@

The variables \(q\) and \(p\) are very specific in Hamiltonian mechanics.

  • \(q = (q_1, q_2, \ldots, q_n)\) are always physical coordinates (positions and angles)

  • \(p = (p_1, p_2, \ldots, p_n)\) are the “conjugate momenta”. That is, each \(p_i\) variable is called a momentum, and \(p_i\) is said to be conjugate to its coordinate \(q_i\).

When put together, the pair \((q, p) \in \mathbb{R}^{2n}\) is said to be canonical. In other words, \((q,p)\) are the canonical variables of the Hamiltonian system described by \(H\). Additionally, the coordinates \((q,p)\) form the system’s phase space. Ignoring some technicalities, this is also often called the state space.

  • \(q\) is always a physical coordinate and therefore is easily understood and interpreted

  • \(p\) is usually not so clear in its physical meaning, but it is related to physical quantities. Given the system’s Lagrangian \(L(q, \dot{q}, t)\), the momenta follow

\[p = \frac{\partial L}{\partial\dot{q}}.\]

Further, the Hamiltonian \(H(q,p,t)\) and Lagrangian \(L(q,\dot{q},t)\) of a given system are actually related to each other.

\[H(q,p,t) = p\cdot\dot{q} - L(q,\dot{q},t)\]

It’s not obvious, but while \(p = L_{\dot{q}}\) directly expresses \(p = p(q,\dot{q},t)\), it actually also always explicitly expresses \(\dot{q} = \dot{q}(q,p,t)\). Therefore, if the Lagrangian \(L(q,\dot{q},t)\) is known, then the Hamiltonian \(H(q,p,t)\) can always be directly calculated.

Hamiltonian invariance@

A special feature of Hamiltonian systems is the following fact.

  • If the Hamiltonian does not explicitly depend on time, then the Hamiltonian is a constant of motion

This is true because of the form of Hamilton’s equations that always result in \(\dot{H} = H_t\), which means that if \(H\) does not depend on \(t\) explicitly then \(\dot{H} = 0\), so the Hamiltonian is constant along trajectories.

\[H(q(t), p(t)) = \text{constant}, \quad \text{if }H = H(q,p)\]

Connection to energy@

A system’s Hamiltonian and Lagrangian are intimately tied up with knowledge about the system’s potential and kinetic energies. In fact, the Hamiltonian and Lagrangian are often written in the coordinate-free notation

\[\begin{split}H &= T+V \\ L &= T-V\end{split}\]

where \(T\) and \(V\) are the kinetic and potential energies, respectively.

Because of this connection with energy and the relationship that the Hamiltonian shares with the Lagrangian, we have the following fact:

  • Any constant can be added to the Hamiltonian without changing the system’s dynamics

This happens because the dynamics depend on the derivatives of \(H\), and not \(H\) itself, but can be shown slightly more rigorously.

\[\begin{split}H_c :\!\!&= H + c \\ &= (T + V) + c \\ &= T + (V + c) \\ &= (p\cdot\dot{q} - L) + c \\ &= p\cdot\dot{q} - (T - V) + c \\ &= p\cdot\dot{q} - (T - (V + c)) \\ &=: p\cdot\dot{q} - L_c \\\end{split}\]

The new system described by \(H_c\) has exactly the same dynamics as that described by \(H\), but the constant \(c\) has been absorbed into the potential energy. Adding a constant to the Hamiltonian essentially amounts to an adjustment of the potential energy’s datum, which as we will see turns out to be useful in some systems.