notes/docs/physics/classical-mechanics/lagrangian-mechanics/lagrange-generalizations.md

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.