Finished Hamiltonian mechanics.

config/en/mkdocs.yaml

@@ -135,7 +135,7 @@ nav:
           - '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': 

docs/en/physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md

@@ -0,0 +1,136 @@
+# 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.

docs/en/physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md

@@ -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
 

docs/en/physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md

@@ -1,2 +0,0 @@
-# Hamilton's equations
-