Added Hamiltonian mechanics.

config/en/mkdocs.yaml

@@ -133,7 +133,9 @@ nav:
         - 'Applications':
           - 'Celestial mechanics': physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md
           - 'Oscillations': physics/mechanics/lagrangian-mechanics/applications/oscillations.md
-#      - 'Hamiltonian mechanics':
+      - 'Hamiltonian mechanics': 
+        - 'Hamiltonian formalism': physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md
+        - "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/hamiltons-equations.md
 #      - 'Relativistic mechanics':
 #    - 'Quantum mechanics':
     - 'Electromagnetism': 

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

@@ -0,0 +1,96 @@
+# Hamiltonian formalism
+
+The Hamiltonian formalism of mechanics is based on the definitions posed by [Lagrangian mechanics](/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism) and the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/). 
+
+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.
+
+## Hamilton's principle
+
+> *Principle 1*: of all the kinematically possible motions that take a mechanical system from one given configuration to another within a time interval $T \subset \mathbb{R}$, the actual motion is the stationary point of the time integral of the Lagrangian $\mathcal{L}$ of the system. Let $S$ be the functional of the trajectories of the system, then
+>
+> $$
+>   S = \int_T \mathcal{L} dt,
+> $$ 
+>
+> has stationary points.
+
+The functional $S$ is often referred to as the action of the system. With this principle the equations of Lagrange can be derived. 
+
+> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange are given by
+>
+> $$
+>   \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) - d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) = 0,
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+??? note "*Proof*:"
+
+    Let the redefined generalized coordinates $\mathbf{q}: (t,a) \mapsto \mathbf{q}(t,a)$ be given by 
+
+    $$
+        \mathbf{q}(t,a) = \mathbf{\hat q}(t) + a \varepsilon(t), 
+    $$
+
+    with $\mathbf{\hat q}: t \mapsto \mathbf{\hat q}(t)$ the generalized coordinates of the system and $\varepsilon: t \mapsto \varepsilon(t)$ a smooth differentiable function. 
+
+    Let $S: a \mapsto S(a)$ be the action of the system and let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian of the system, according to Hamilton's principle 
+
+    $$
+        S(a) = \int_T \mathcal{L}(\mathbf{q}, \mathbf{q'})dt,
+    $$
+
+    for all $a \in \mathbb{R}$. To determine the stationary points we must have that $S'(0) = 0$. We have that $S'$ is given by
+
+    $$
+    \begin{align*}
+        S'(a) &= \int_T \partial_a \mathcal{L}(\mathbf{q}, \mathbf{q'})dt, \\
+            &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \partial_a q_j + \partial_{q_j'} \mathcal{L} \partial_a q_j'\bigg)dt, \\
+            &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) + \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j\bigg)dt. \\
+    \end{align*}
+    $$
+
+    Partial integration may be used for the second part:
+
+    $$
+    \begin{align*}
+        \int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt &= \Big[\partial_{q_j'} \mathcal{L} \partial_a q_j \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt, \\
+        &= \Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
+    \end{align*}
+    $$
+
+    Choose $\varepsilon_j$ such that
+
+    $$
+        \Big[\partial_{q_j'} \mathcal{L} \varepsilon_j(t) \Big]_T = 0.
+    $$
+
+    Obtains
+
+    $$
+        \int_T \partial_{q_j'} \mathcal{L} \partial_a \partial_t q_j dt = - \int_T \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})dt.
+    $$
+
+    The general expression of $S'$ may now be given by
+
+    $$
+    \begin{align*}
+        S'(a) &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \partial_a q_j d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
+            &= \int_T \sum_{j=1}^f \bigg(\partial_{q_j} \mathcal{L} \varepsilon_j(t) - \varepsilon_j(t) d_t (\partial_{q_j'} \mathcal{L})\bigg)dt, \\
+            &= \sum_{j=1}^f \int_T  \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt.
+    \end{align*}
+    $$
+
+    Then
+
+    $$
+        S'(0) = \sum_{j=1}^f \int_T  \varepsilon_j(t) \Big(\partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L})\Big)dt = 0,
+    $$
+
+    since $\varepsilon_j$ can be chosen arbitrary this implies that 
+
+    $$
+        \partial_{q_j} \mathcal{L} - d_t (\partial_{q_j'} \mathcal{L}) = 0.
+    $$

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

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

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

@@ -1,6 +1,6 @@
 # Lagrangian formalism
 
-The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/newtonian-mechanics/newtonian-formalism.md).
+The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/newtonian-formalism/).
 
 ## Configuration of a system
 
@@ -65,7 +65,7 @@ for all $t \in \mathbb{R}$ (inexplicitly).
 > with
 >
 > $$
->   a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}),
+>   a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \Big\langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}) \Big\rangle,
 > $$
 >
 > for all $t \in \mathbb{R}$.
@@ -73,4 +73,3 @@ for all $t \in \mathbb{R}$ (inexplicitly).
 ??? note "*Proof*:"
 
     Will be added later.
-