Equations of Hamilton

The Hamiltonian

Definition 1: let \(\mathcal{L}: (\mathbf{q},\mathbf{q}',t) \mapsto \mathcal{L}(\mathbf{q},\mathbf{q}',t)\) be the Lagrangian of the system, suppose that the generalized momenta \(\mathbf{p}\) are defined in terms of the active variables \(\mathbf{q}'\) and the passive variables \((\mathbf{q},t)\) such that

\[ \mathbf{p} = \nabla_{\mathbf{q}'}\mathcal{L}(\mathbf{q},\mathbf{q}',t), \]

for all \(t \in \mathbb{R}\).

We may now pose that there exists a function that meets the inverse, which can be obtained with Legendre transforms.

Theorem 1: there exists a function \(\mathcal{H}: (\mathbf{q},\mathbf{p},t) \mapsto \mathcal{H}(\mathbf{q},\mathbf{p},t)\) such that

\[ \mathbf{q}' = \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \]

for all \(t \in \mathbb{R}\). Where \(\mathcal{H}\) is the Hamiltonian of the system and is related to the Lagrangian \(\mathcal{L}\) by

\[ \mathcal{H}(\mathbf{q},\mathbf{p},t) = \langle \mathbf{q'}, \mathbf{p} \rangle - \mathcal{L}(\mathbf{q},\mathbf{q}',t), \]

for all \(t \in \mathbb{R}\) with \(\mathcal{L}\) and \(\mathcal{H}\) the Legendre transforms of each other.

Proof:

Will be added later.

The equations of Hamilton

Corollary 1: the partial derivatives of \(\mathcal{L}\) and \(\mathcal{H}\) with respect to the passive variables are related by

\[ \begin{align*} \nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \nabla_{\mathbf{q}} \mathcal{L}(\mathbf{q},\mathbf{q}',t), \\ \partial_t \mathcal{H}(\mathbf{q},\mathbf{p},t) &= - \partial_t \mathcal{L}(\mathbf{q},\mathbf{q}',t), \end{align*} \]

for all \(t \in \mathbb{R}\).

Proof:

Will be added later.

Obtaining the equations of Hamilton

\[ \begin{align*} \mathbf{p}' &= -\nabla_{\mathbf{q}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \\ \mathbf{q}' &= \nabla_{\mathbf{p}} \mathcal{H}(\mathbf{q},\mathbf{p},t), \end{align*} \]

for all \(t \in \mathbb{R}\).

Proposition 1: when the Hamiltonian \(\mathcal{H}\) has no explicit time dependence it is a constant of motion.

Proof:

Will be added later.

To put it differently; a Hamiltonian of a conservative autonomous system is conserved.

Theorem 2: for conservative autonomous systems, the Hamiltonian \(\mathcal{H}\) may be expressed as

\[ \mathcal{H}(\mathbf{q},\mathbf{p}) = T(\mathbf{q},\mathbf{p}) + V(\mathbf{q}), \]

for all \(t \in \mathbb{R}\) with \(T: (\mathbf{q},\mathbf{p}) \mapsto T(\mathbf{q},\mathbf{p})\) and \(V: \mathbf{q} \mapsto V(\mathbf{q})\) the kinetic and potential energy of the system.

Proof:

Will be added later.

It may be observed that the Hamiltonian \(\mathcal{H}\) and generalised energy \(h\) are identical. Note however that \(\mathcal{H}\) must be expressed in \((\mathbf{q},\mathbf{p},t)\) which is not the case for \(h\).

Proposition 2: a coordinate \(q_j\) is cyclic if

\[ \partial_{q_j} \mathcal{H}(\mathbf{q},\mathbf{p},t) = 0, \]

for all \(t \in \mathbb{R}\).

Proof:

Will be added later.

Proposition 3: the Hamiltonian is seperable if there exists two mutually independent subsystems.

Proof:

Will be added later.

Poisson brackets

Definition 2: let \(G: (\mathbf{q},\mathbf{p},t) \mapsto G(\mathbf{q},\mathbf{p},t)\) be an arbitrary observable, its time derivative may be given by

\[ \begin{align*} d_t G(\mathbf{q},\mathbf{p},t) &= \sum_{j=1}^f \Big(\partial_{q_j} G q_j' + \partial_{p_j} G p_j' \Big) + \partial_t G, \\ &= \sum_{j=1}^f \Big(\partial_{q_j} G \partial_{p_j} \mathcal{H} - \partial_{p_j} G \partial_{q_j} \mathcal{H} \Big) + \partial_t G, \\ &\overset{\mathrm{def}}= \{G, \mathcal{H}\} + \partial_t G. \end{align*} \]

for all \(t \in \mathbb{R}\) with \(\mathcal{H}\) the Hamiltonian and \(\{G, \mathcal{H}\}\) the Poisson bracket of \(G\) and \(\mathcal{H}\).

The Poisson bracket may simplify expressions; it has distinct properties that are true for any observables. The following theorem demonstrates the usefulness even more.

Theorem 3: let \(f: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)\) and \(g: (\mathbf{q}, \mathbf{p}, t) \mapsto f(\mathbf{q}, \mathbf{p}, t)\) be two integrals of Hamilton's equations given by

\[ \begin{align*} f(\mathbf{q}, \mathbf{p}, t) = c_1, \\ g(\mathbf{q}, \mathbf{p}, t) = c_2, \end{align*} \]

for all \(t \in \mathbb{R}\) with \(c_{1,2} \in \mathbb{R}\). Then

\[ \{f,g\} = c_3 \]

with \(c_3 \in \mathbb{R}\) for all \(t \in \mathbb{R}\).

Proof:

Will be added later.