Original notes on variational principles in mechanics.
Paths to Lagrange's Equation, all assuming ideal constraints:
\[\begin{aligned} \text{Principle of Virtual Work} \Rightarrow \text{Lagrange-d'Alembert's Principle} &\Rightarrow \text{Lagrange's Equation} \\ \text{Hamilton's Principle} &\Rightarrow \text{Lagrange's Equation} \end{aligned}\] Lagrange's Equation, Hamilton's Equations, and Newton's Second Law are equivalent.
For any virtual displacement of a system at equilibrium, the virtual work of external forces equals the virtual work of deformation.
\[ \delta V = \delta W_\text{external} \]
In other words, equilibrium condition is equivalent to total virtual work being zero for all virtual displacements:
\[ Q_j = 0~(j = 1, \cdots, s) \Leftrightarrow \sum_{i=1}^{n} \mathbf{F}_i \cdot \delta \mathbf{r}_i = 0 \]
For a system of \(n\) particles, where \(\mathbf{F}_i\) is the total non-constraint force on the \(i\)-th particle:
\[ \sum_{i=1}^{n} (\mathbf{F}_i - m_i \mathbf{a}_i) \cdot \delta \mathbf{r}_i = 0 \]
For a system of \(n\) particles in a 3-dimensional space, its kinetic energy: \( T = \sum_{i=1}^n \frac{1}{2} m_i \mathbf{v}_i^2 \).
The natural coordinates are subject to \(k\) ideal constraints (never do work on the system): \(f_i(x_1, \cdots, x_{3n}, t) = 0\).
Define components of non-constraint forces: \( Q_j = \sum_{i=1}^{n} \mathbf{F}_i \cdot \frac{\partial \mathbf{r}_i}{\partial x_j} \), where \( \mathbf{F}_i \) is the total force except constraint forces on the \(i\)-th particle.
Lagrange's equation of the first kind: For \(j = 1, \cdots, 3n\), and Lagrangian coefficients \( \lambda_i \), \[ \frac{\text{d}}{\text{d}t} \frac{\partial T}{\partial \dot{x}_j} - \frac{\partial T}{\partial x_j} = Q_j + \sum_{i=1}^k \lambda_i \frac{\partial f_i}{\partial x_j} \]
The system has degree of freedom: \(s = 3n - k\). With independent generalized coordinates \( q_j \), \( j = 1, \cdots, s \), define generalized forces: \( Q_j = \sum_{i = 1}^{n} \mathbf{F}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j} \).
Lagrange's equation of the second kind: For \( j = 1, \cdots, s \), \[ \frac{\text{d}}{\text{d}t} \frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j} = Q_j \]
Lagrangian is kinetic energy plus virtual work of external forces: \( L = T + W \). If all forces are conservative and have a total potential: \(V(\mathbf{r}_1, \cdots, \mathbf{r}_n) = V(q_1, \cdots, q_s, t)\), then the Lagrangian equals kinetic energy minus potential energy: \( L = T - V \), and Lagrange's equation is simply: \[ \frac{\text{d}}{\text{d}t} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = 0 \]
Action integral is the time integral of Lagrangian: \( S = \int_{t_1}^{t_2} L~\text{d}t \).
Of all variations, the dynamical path of a system will lead to a stationary value of the action integral: \[ \delta S = \int_{t_1}^{t_2} \left(\delta T - \delta V + \overline{\delta W_\text{nc}}\right)~\text{d}t = 0 \] Here \( W_\text{nc} \) is the work done by non-conservative forces.
For a system of \(s\) degress of freedom, with independent generalized coordinates \( q_i \) and generalized momentum \( p_i = \frac{\partial L}{\partial \dot{q}_i} \), Hamiltonian is the Legendre transformation of the Lagrangian: \( H(\mathbf{q}, \mathbf{p}, t) = \sum_{i=1}^{s} p_i \dot{q}_i - L \).
\[\begin{aligned} \dot{q}_i &= \frac{\partial H}{\partial p_i} \\ \dot{p}_i &= - \frac{\partial H}{\partial q_i} \end{aligned}\]
Hamilton's Principle implies Minimum Total Potential Energy Principle and Minimum Total Complementary Energy Principle.
Total potential energy is deformation energy minus the work of non-conservative forces: \( \Phi = V - W_\text{nc} \).
\[ \frac{\partial \Phi}{\partial q_i} = 0 \]
Total complementary energy is the Legendre transformation of total potential energy, with respect to conjugate pairs of generalized coordinates and forces: \( \Psi = q_s Q_s - V \).
\[ \frac{\partial \Psi}{\partial Q_i} = 0 \]