3.8 KiB
The equations of Lagrange
Principle of virtual work
Definition 1: a virtual displacement is a displacement at a fixed moment in time that is consistent with the constraints at that moment.
The following principle addresses the problem that the constraint forces are generally unknown.
Principle 1: let
\mathbf{\delta x}_i \in \mathbb{R}^m
be a virtual displacement and let\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})
be the total force excluding the constraint forces. Then
\sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}) - m_i \mathbf{x}_i''(\mathbf{q}), \mathbf{\delta x}_i \Big\rangle = 0,
is true for sklerenomic constraints and all
t \in \mathbb{R}
.
Which implies that the constraint forces do not do any (net) virtual work.
The equations of Lagrange
Theorem 1: let
T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})
be the kinetic energy of the system. For holonomic constraints we have that
d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}') \Big) - \partial_{q_j} T(\mathbf{q},\mathbf{q}') = Q_j(\mathbf{q}),
for all
t \in \mathbb{R}
. WithQ_j: \mathbf{q} \mapsto Q_j(\mathbf{q})
the generalized forces of type I given by
Q_j(\mathbf{q}) = \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}), \partial_j \mathbf{x}_i(\mathbf{q}) \Big\rangle,
for all
t \in \mathbb{R}
with\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})
the total force excluding the constraint forces.
??? note "Proof:"
Will be added later.
Obtaining the equations of Lagrange. Note that the position of each point mass \mathbf{x}_i
is defined in the Lagrangian formalism.
Conservative systems
For conservative systems we may express the force \mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})
in terms of a potential energy V: X \mapsto V(X)
by
\mathbf{F}_i(\mathbf{q}) = -\nabla_i V(X),
for X: \mathbf{q} \mapsto X(\mathbf{q}) \overset{\mathrm{def}}= \{\mathbf{x}_i(\mathbf{q})\}_{i=1}^n
.
Lemma 1: for a conservative holonomic system the generalized forces of type I
Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})
may be expressed in terms of the potential energyV: \mathbf{q} \mapsto V(\mathbf{q})
by
Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}),
for all
t \in \mathbb{R}
.
??? note "Proof:"
Will be added later.
The equation of Lagrange may now be rewritten, which obtains the following lemma.
Lemma 2: let
T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})
andV: \mathbf{q} \mapsto V(\mathbf{q})
be the kinetic and potential energy of the system. The Lagrange equations for conservative systems are given by
d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}')\Big) - \partial_{q_j}T(\mathbf{q},\mathbf{q}') = - \partial_{q_j} V(\mathbf{q}),
for all
t \in \mathbb{R}
??? note "Proof:"
Will be added later.
Definition 2: let
T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})
andV: \mathbf{q} \mapsto V(\mathbf{q})
be the kinetic and potential energy of the system. The Lagrangian\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})
is defined as
\mathcal{L}(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') - V(\mathbf{q}),
for all
t \in \mathbb{R}
.
With this definition we may write the Lagrange equations in a more formal way.
Theorem 2: let
\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})
be the Lagrangian, the equations of Lagrange for conservative holonomic systems are given by
d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) - \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
for all
t \in \mathbb{R}
.
??? note "Proof:"
Will be added later.