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Finished Hamiltonian mechanics.

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Luc Bijl 2024-04-04 11:21:54 +02:00
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- 'Oscillations': physics/mechanics/lagrangian-mechanics/applications/oscillations.md
- 'Hamiltonian mechanics':
- 'Hamiltonian formalism': physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md
- "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md
- "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md
# - 'Relativistic mechanics':
# - 'Quantum mechanics':
- 'Electromagnetism':

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# 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 $t \in \mathbb{R}$ 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.
??? note "*Proof*:"
Will be added later.
It may be observed that the Hamiltonian $\mathcal{H}$ and [generalised energy](/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations/#the-generalized-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 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 $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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@ -4,7 +4,7 @@ The Hamiltonian formalism of mechanics is based on the definitions posed by [Lag
Where the Lagrangian formalism used the [principle of virtual work](/en/physics/mechanics/lagrangian-mechanics/lagrange-equations/#principle-of-virtual-work) to derive the Lagrangian equations of motion, the Hamiltonian formalism will derive the Lagrangian equations with the stationary action principle. A derivative of Fermat's principle of least time.
In Hamilton's formulation the principle is referred to as Hamilton's principle.
In Hamilton's formulation the stationary action principle is referred to as Hamilton's principle.
## Hamilton's principle

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# Hamilton's equations