Finished Lagrangian mechanics.

config/en/mkdocs.yaml

@@ -130,6 +130,9 @@ nav:
         - 'Lagrangian formalism': physics/mechanics/lagrangian-mechanics/lagrangian-formalism.md
         - 'Lagrange equations': physics/mechanics/lagrangian-mechanics/lagrange-equations.md
         - 'Lagrange generalizations': physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md
+        - 'Applications':
+          - 'Celestial mechanics': physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md
+          - 'Oscillations': physics/mechanics/lagrangian-mechanics/applications/oscillations.md
 #      - 'Hamiltonian mechanics':
 #      - 'Relativistic mechanics':
 #    - 'Quantum mechanics':

docs/en/physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md

@@ -0,0 +1 @@
+# Celestial mechanics

docs/en/physics/mechanics/lagrangian-mechanics/applications/oscillations.md

@@ -0,0 +1 @@
+# Oscillations

docs/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md

@@ -1,2 +1,96 @@
 # Lagrange generalizations
 
+## The generalized momentum and force
+
+> *Definition 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized momentum** $p_j: (\mathbf{q}, \mathbf{q}') \mapsto p_j(\mathbf{q},\mathbf{q}')$ is defined as
+>
+> $$
+>   p_j(\mathbf{q},\mathbf{q}') = \partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}), 
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+The generalized momentum may also be referred to as the canonical or conjugated momentum. Recall that $j \in \mathbb{N}[j\leq f]$. 
+
+> *Definition 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized force of type II** $F_j: (\mathbf{q}, \mathbf{q}') \mapsto F_j(\mathbf{q},\mathbf{q}')$ is defined as
+>
+> $$
+>   F_j(\mathbf{q},\mathbf{q}') = \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'})
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+We may also write $\mathbf{p} = \{p_j\}_{j=1}^f$ and $\mathbf{F} = \{F_j\}_{j=1}^f$. 
+
+## The generalized energy
+
+> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the generalized energy $h: (\mathbf{q}, \mathbf{q'},\mathbf{p}) \mapsto h(\mathbf{q}, \mathbf{q'},\mathbf{p})$ is given by
+>
+> $$
+>   h(\mathbf{q}, \mathbf{q'}, \mathbf{p}) = \sum_{j=1}^f \big(p_j q_j' \big) - \mathcal{L}(\mathbf{q}, \mathbf{q'}), 
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+A generalization of the concept of energy.
+
+* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'},t) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'},t)$ is explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'},t) \neq 0$ and the generalized energy $h$ is not conserved.
+* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is not explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0$ and the generalized energy $h$ is conserved. 
+
+> *Theorem 2*: for autonomous systems with only conservative forces the generalized energy $h: (\mathbf{q}, \mathbf{q'}) \mapsto h(\mathbf{q}, \mathbf{q'})$ is conserved and is given by
+>
+> $$
+>   h(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') + V(\mathbf{q}) \overset{\mathrm{def}}= E,
+> $$
+>
+> for all $t \in \mathbb{R}$ with $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system and $E \in \mathbb{R}$ the total energy of the system.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+In this case the generalized energy $h$ is conserved and is equal to the total energy $E$ of the system.
+
+## Conservation of generalized momentum
+
+> *Definition 3*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, a coordinate $q_j$ is **cyclic** if
+>
+> $$
+>   \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+Therefore the Lagrangian is independent of a cyclic coordinate. 
+
+> *Proposition 1*: the generalized momentum $p_j$ corresponding to a cyclic coordinate $q_j$ is conserved.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Seperable systems
+
+> *Proposition 2*: the Lagrangian is seperable if there exists two mutually independent subsystems. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Obtaining a decoupled set of partial differential equations.
+
+## Invariances
+
+> *Proposition 3*: the Lagrangian is invariant for Gauge transformations and therefore **not unique**. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+There can exist multiple Lagrangians that may lead to the same equation of motion.
+
+According to the theorem of Noether, the invariance of a closed system with respect to continuous transformations implies that corresponding conservation laws exist.

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](../newtonian-mechanics/newtonian-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).
 
 ## Configuration of a system