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 \(t \in \mathbb{R}\). With \(Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})\) 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.

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 energy \(V: \mathbf{q} \mapsto V(\mathbf{q})\) by

\[ Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}), \]

for all \(t \in \mathbb{R}\).

Proof:

Will be added later.

The equation of Lagrange may now be rewritten, which obtains the following lemma.

Lemma 2: let \(T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})\) and \(V: \mathbf{q} \mapsto V(\mathbf{q})\) 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 \(t \in \mathbb{R}\)

Proof:

Will be added later.

Definition 2: let \(T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})\) and \(V: \mathbf{q} \mapsto V(\mathbf{q})\) 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 \(t \in \mathbb{R}\).

Proof:

Will be added later.