3.8 KiB
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 momentump_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 IIF_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 energyh: (\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 energyh
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 energyh
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}
withT: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})
andV: \mathbf{q} \mapsto V(\mathbf{q})
the kinetic and potential energy of the system andE \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 coordinateq_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 coordinateq_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.