4.6 KiB
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.
??? note "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}
.
??? note "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.
??? note "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}
withT: (\mathbf{q},\mathbf{p}) \mapsto T(\mathbf{q},\mathbf{p})
andV: \mathbf{q} \mapsto V(\mathbf{q})
the kinetic and potential energy of the system.
??? note "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}
.
??? note "Proof:"
Will be added later.
Proposition 3: the Hamiltonian is seperable if there exists two mutually independent subsystems.
??? note "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 ofG
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)
andg: (\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}
withc_{1,2} \in \mathbb{R}
. Then
{f,g} = c_3
with
c_3 \in \mathbb{R}
for allt \in \mathbb{R}
.
??? note "Proof:"
Will be added later.