Examples: Fermat's principle in geometrical optics; variational method in quantum mechanics; the Einstein–Hilbert action and Palatini variation in general relativity; principle of least action in electromagnetism;

## Classical 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.

On Encyclopedia of Math, also includes principle of least compulsion (Gauss' principle), principle of least curvature (Hertz' principle), Chetaev's principle of maximum wor;, Jourdain's principle; Lagrange's principle of stationary action, Jacobi's principle of stationary action.

### Principle of Virtual Work

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:

$$Q_j = 0~(j = 1, \cdots, s) \Leftrightarrow \sum_{i=1}^{n} \mathbf{F}_i \cdot \delta \mathbf{r}_i = 0$$

### Lagrange-d'Alembert's Principle

For a system of $n$ particles, where $\mathbf{F}_i$ is the total non-constraint force on the i-th particle:

$$\sum_{i=1}^{n} (\mathbf{F}_i - m_i \mathbf{a}_i) \cdot \delta \mathbf{r}_i = 0$$

### Lagrange's Equation

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: $$\frac{\text{d}}{\text{d}t} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = 0$$

### Hamilton's Principle

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.

### Hamilton's Equations

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}

## Mechanics of Materials, Elasticity

Hamilton's Principle implies Minimum Total Potential Energy Principle and Minimum Total Complementary Energy Principle.

### Minimum Total Potential Energy Principle

Total potential energy is deformation energy minus the work of non-conservative forces: $\Phi = V - W_\text{nc}$.

$$\frac{\partial \Phi}{\partial q_i} = 0$$

### Minimum Total Complementary Energy Principle

Total complementary energy is the Legendre transformation of total potential energy, with respect to conjugate pairs of generalized coordinates and forces: $\Psi = q_s Q_s - V$.

$$\frac{\partial \Psi}{\partial Q_i} = 0$$