5.5. Properties of the Lagrangian approach to classical mechanics

5.5.1. Kinetic energy \(K\) and potential function \(U\)

Kinetic energy of each component is a symmetric function of the velocity of its reference point \(P\) and its angular velocity \(\vec{\omega}\), as shown as an example by the expression of the kinetic energy of a rigid body,

\[\begin{split}\begin{aligned} K_P & = \dfrac{1}{2} \vec{v}_P \cdot \vec{Q} + \dfrac{1}{2} \vec{\omega} \cdot \vec{L}_P \\ & = \dfrac{1}{2} \vec{v}_P \cdot \left[ m \vec{v}_P + \mathbb{S}_P \cdot \vec{\omega} \right] + \dfrac{1}{2} \vec{\omega} \cdot \left[ \mathbb{S}_P^T \cdot \vec{v}_P + \mathbb{I}_P \cdot \vec{\omega} \right] \ , \end{aligned}\end{split}\]

as the tensor of inertia \(\mathbb{I}_P\) is symmetric, i.e. \(\vec{a} \cdot \mathbb{I}_P \cdot \vec{b} = \vec{b} \cdot \mathbb{I}_P \cdot \vec{a}\), \(\forall \vec{a}, \vec{b}\).

todo Treat continuous deformable systems.

Kinetic energy of a system, with no explicit dependence of time of the degrees of freedom is a symmetric positive definite quadratic function.

\[K = \dfrac{1}{2} \sum_{ik} M_{ik}\left(q^l(t)\right) \dot{q}^i \dot{q}^k = \dfrac{1}{2} \dot{\mathbf{q}}^T \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} \ ,\]

with \(M_{ik} = M_{ki}\), and \(K \ge 0\).

Kinetic energy symmetric quadratic form of generalized coordinates.

If degrees of freedom of the system can be written as functions of the generalized coordinates \(q^k(t)\), with no explicit dependence on time \(t\),

\[\vec{r}_P\left( q^k(t)\right) \qquad , \qquad \mathbb{R}\left( q^k(t) \right) \ ,\]

velocities and angular velocities becomes

\[\begin{split}\begin{aligned} \vec{v}_P\left( \dot{q}^k(t), q^k(t) \right) & = \dot{q}^k \dfrac{\partial \vec{r}_P}{\partial q^k}\left( q^k(t) \right) \\ \vec{\omega}_P\left( \dot{q}^k(t), q^k(t) \right) & = \dot{q}^k \dfrac{\partial \vec{\theta}_P}{\partial q^k}\left( q^k(t) \right) \\ \end{aligned}\end{split}\]

The kinetic energy of the system is the sum of the kinetic energy of its parts and thus

\[\begin{split}\begin{aligned} K = \sum_P K_P & = \sum_P \left\{ \dfrac{1}{2} \vec{v}_P \cdot \left[ m \vec{v}_P + \mathbb{S}_P \cdot \vec{\omega} \right] + \dfrac{1}{2} \vec{\omega} \cdot \left[ \mathbb{S}_P^T \cdot \vec{v}_P + \mathbb{I}_P \cdot \vec{\omega} \right] \right\} = \\ & = \sum_P \left\{ \dfrac{1}{2} \sum_i \dot{q}^i \partial_{q^i} \vec{r}_P \cdot \left[ m \sum_k \dot{q}^k \partial_{q^k} \vec{r}_P + \mathbb{S}_P \cdot \sum_k \dot{q}^k \partial_{q^k} \vec{\theta}_P \right] \right. + \\ & \qquad \left. + \dfrac{1}{2} \sum_i \dot{q}^i \partial_{q^i} \vec{\theta}_P \cdot \left[ \mathbb{S}_P^T \cdot \sum_k \dot{q}^k \partial_{q^k} \vec{r}_P + \mathbb{I}_P \cdot \sum_k \dot{q}^k \partial_{q^k} \vec{\theta}_P \right] \right\} = \\ & = \dfrac{1}{2} \sum_{i,k} \sum_P \left\{ m \partial_i \vec{r}_P \cdot \partial_k \vec{r}_P + \partial_i \vec{r}_P \cdot \mathbb{S}_P \cdot \partial_k \vec{\theta}_P + \partial_i \vec{\theta}_P \cdot \mathbb{S}_P^T \cdot \partial_k \vec{r}_P + \partial_i \vec{\theta}_P \cdot \mathbb{I}_P \cdot \partial_k \vec{\theta}_P \right\} \dot{q}^i \dot{q}^k = \\ & = \dfrac{1}{2} \sum_{ik} M_{ik} \dot{q}^i \, \dot{q}^k \ , \end{aligned}\end{split}\]

with the coefficients \(M_{ik}(q^l(t))\) symmetric w.r.t. a swap of the indices \(i\), \(k\) by definition.

Potential function is a function of generalized coordinates only, \(U(\mathbf{q})\).

5.5.2. Lagrange equations

Lagrangian function is

\[L(\dot{\mathbf{q}}, \mathbf{q}) = K(\dot{\mathbf{q}}, \mathbf{q}) + U(\mathbf{q}) = \dfrac{1}{2} \dot{\mathbf{q}}^T \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} + U\left(\mathbf{q} \right)\]

5.5.2.1. Lagrange equations of the II kind

Lagrange equations of the II kind read

\[\begin{split}\begin{aligned} \mathbf{Q}_{\mathbf{q}} & = \dfrac{d}{dt} \left( \partial_{\dot{\mathbf{q}}} L \right) - \partial_{\mathbf{q}} L = \\ & = \dfrac{d}{dt} \left( \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} \right) - \partial_{\mathbf{q}} \left( \dfrac{1}{2} \dot{\mathbf{q}}^T \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} + U (\mathbf{q}) \right) \ , \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} Q_i & = \dfrac{d}{dt} \left( M_{ij}(q^k) \dot{q}^j \right) - \partial_{q^i} \left( \dfrac{1}{2} \dot{q}^a M_{ab}(q^k) \dot{q}^b + U(q^k) \right) = \\ & = \partial_{q^k} M_{ij}(q^l) \dot{q}^k \dot{q}^j + M_{ij} \ddot{q}^j - \dfrac{1}{2} \partial_{q^i} M_{kj}(q^l) \dot{q}^k \dot{q}^j - \partial_{q^i} U(q^l) \\ \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} M_{ij} (q^l) \ddot{q}^j - \partial_{q^i} U(q^l) + \partial_{q^k} M_{ij}(q^l) \dot{q}^k \dot{q}^j - \frac{1}{2} \partial_{q^i} M_{kj}(q^l) \dot{q}^k \dot{q}^j & = Q_i \\ \mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} - \nabla_{\mathbf{q}} U(\mathbf{q}) + \dot{\mathbf{q}}^T \nabla_{\mathbf{q}} \left( \mathbf{M} \dot{\mathbf{q}} \right) - \frac{1}{2} \nabla_{\mathbf{q}} \left( \dot{\mathbf{q}}^T \mathbf{M} \dot{\mathbf{q}} \right) & = \mathbf{Q} \end{aligned}\end{split}\]

Equilibrium conditions correspond to steady solutions of the equations of motion,

\[-\nabla_{\mathbf{q}} U (\overline{\mathbf{q}}) = \mathbf{Q}(\overline{\mathbf{q}}) \ .\]

Linearized system around a (stable) equilibrium follows from the approximation

\[\begin{split}\begin{aligned} \mathbf{q} & = \overline{\mathbf{q}} + \mathbf{q}' \\ \dot{\mathbf{q}} & = \dot{\mathbf{q}}' \\ \ddot{\mathbf{q}} & = \ddot{\mathbf{q}}' \ , \end{aligned}\end{split}\]

and neglecting higher-order contributions in \(\mathbf{q}'\),

\[\begin{split}\begin{aligned} Q_i (\mathbf{q}) & = M_{ij}(\mathbf{q}) \ddot{q}^j - \partial_i U(\mathbf{q}) + \dot{q}^k \partial_k M_{ij}(\mathbf{q}) \dot{q}^j - \frac{1}{2} \partial_i M_{jk}(\mathbf{q}) \dot{q}^j \dot{q}^k \\ Q_i(\overline{\mathbf{q}}) + {q^{k}}' \partial_{q^k} Q_i(\overline{\mathbf{q}}) & \sim M_{ij}(\overline{\mathbf{q}}) \ddot{q}^j - \partial_i U(\overline{\mathbf{q}}) - q^{j'} \partial_{ij} U(\overline{\mathbf{q}}) \\ \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} M_{ij}(\overline{\mathbf{q}}) \ddot{q}'_j + \left[ - \partial_{ij} U(\overline{\mathbf{q}}) - \partial_j Q_i(\overline{\mathbf{q}}) \right] q'_j & = 0 \\ \mathbf{M}(\overline{\mathbf{q}}) \ddot{\mathbf{q}}' + \left[ - \nabla \nabla U(\overline{\mathbf{q}}) \right] \mathbf{q}' & = \nabla \mathbf{Q}^T (\overline{\mathbf{q}}) \mathbf{q}' \\ \mathbf{M}(\overline{\mathbf{q}}) \ddot{\mathbf{q}}' + \mathbf{K}(\overline{\mathbf{q}}) \mathbf{q}' & = \nabla \mathbf{Q}^T(\overline{\mathbf{q}})\mathbf{q}' \ . \end{aligned}\end{split}\]

Matrices are symmetric. Mass matrix is symmetric by construction, as already proved. Stiffness matrix is symmetric as well, due to Schwarz’s theorem, as its the Hessian of a scalar function, \(\left[ \nabla \nabla U \right]_{ij} = \partial_{ij} U\)

Matrices are (semi)-definite positive. Mass matrix is definite positive by definition, \(\mathbf{M} > 0\), as already proved above as the kinetic energy is a non-negative scalar physical quantity. Stiffness matrix is definite positive if the equilibrium \(\overline{\mathbf{q}}\) is stable: \(\nabla \nabla U(\overline{\mathbf{q}}) < 0\) implies \(\mathbf{K} > 0\).

Spectrum of a stable system. todo add a link to structure mechanics in continuum mechanics, and modal approach

5.5.2.2. Lagrange equations of the I kind

Recasting the expression of the dynamical equations of an “uncostrained” system (or with some constraints treated implicitly with Lagrange equations of II kind),

\[\mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} = \mathbf{f}(\mathbf{q}, \dot{\mathbf{q}}) \ ,\]

and then adding constraints - at least holonomic or semi-linear non-holonomic constraints - and the costraint reactions, the system becomes

\[\begin{split}\begin{aligned} & \mathbf{M}(\mathbf{q}) \ddot{\mathbf{q}} = \mathbf{f}(\mathbf{q}, \dot{\mathbf{q}}) + \nabla_\mathbf{q} \mathbf{g}(\mathbf{q}) \, \mathbf{s} \\ & \mathbf{g}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{0} \end{aligned}\end{split}\]

with either \(\mathbf{g}(\mathbf{q}) = \mathbf{0}\) for holonomic time-independent constraints, or \(\mathbf{g}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{a}(\mathbf{q}) \dot{\mathbf{q}} = \mathbf{0}\) for semi-linear non-holonomic time-independent constraints, with the forcing term \(\mathbf{f}\) containing the non-conservative forces \(\mathbf{Q}\) (or at least contributions not included in the potential \(U(\mathbf{q})\)), the conservative forces \(\nabla_{\mathbf{q}} U(\mathbf{q})\), and the (quadratic! Important for neglecting them in linearization) contributions from time derivative of kinetic energy,

\[\mathbf{f}(\mathbf{q}, \dot{\mathbf{q}}) = \mathbf{Q} + \nabla_{\mathbf{q}} U(\mathbf{q}) - \dot{\mathbf{q}}^T \nabla_{\mathbf{q}} \left( \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} \right) + \frac{1}{2} \nabla_{\mathbf{q}} \left( \dot{\mathbf{q}}^T \mathbf{M}(\mathbf{q}) \dot{\mathbf{q}} \right) \ .\]

Linearized equations become

\[\begin{split}\begin{aligned} & \mathbf{M}(\overline{\mathbf{q}}) \ddot{\mathbf{q}}' + \mathbf{K}(\overline{\mathbf{q}}) \mathbf{q}' = \mathbf{Q}' + \nabla_{\mathbf{q}} \mathbf{g}(\overline{\mathbf{q}}) \mathbf{s}' + \mathbf{q}' \nabla_{\mathbf{q}} \nabla_{\mathbf{q}} \mathbf{g}(\overline{\mathbf{q}}) \mathbf{s} \\ & \nabla_{\dot{\mathbf{q}}}^T \mathbf{g}(\overline{\mathbf{q}}, \mathbf{0}) \, \dot{\mathbf{q}}' + \nabla_{\mathbf{q}}^T \mathbf{g}(\overline{\mathbf{q}}, \mathbf{0}) \, \mathbf{q}' = \mathbf{0} \end{aligned}\end{split}\]