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 \(t \in \mathbb{R}\).
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.
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.
Proof:
Will be added later.
Seperable systems
Proposition 2: the Lagrangian is seperable if there exists two mutually independent subsystems.
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.
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.