diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 4ca8e1e..a21ded8 100755 --- a/config/en/mkdocs.yaml +++ b/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': diff --git a/docs/en/physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md b/docs/en/physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md new file mode 100644 index 0000000..60828c0 --- /dev/null +++ b/docs/en/physics/mechanics/lagrangian-mechanics/applications/celestial-mechanics.md @@ -0,0 +1 @@ +# Celestial mechanics \ No newline at end of file diff --git a/docs/en/physics/mechanics/lagrangian-mechanics/applications/oscillations.md b/docs/en/physics/mechanics/lagrangian-mechanics/applications/oscillations.md new file mode 100644 index 0000000..0443df6 --- /dev/null +++ b/docs/en/physics/mechanics/lagrangian-mechanics/applications/oscillations.md @@ -0,0 +1 @@ +# Oscillations \ No newline at end of file diff --git a/docs/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md b/docs/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md index 69ced4e..67cd1be 100644 --- a/docs/en/physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md +++ b/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. \ No newline at end of file diff --git a/docs/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism.md b/docs/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism.md index eac434a..a08ad58 100644 --- a/docs/en/physics/mechanics/lagrangian-mechanics/lagrangian-formalism.md +++ b/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