Variational Principles in Mechanics and Physics

"Nature prefers simplicity"—this aphorism is embodied in the principle of least action, one of the deepest principles of physics. Instead of "solving" the equations of motion step by step for each point of a trajectory, nature "chooses" the entire trajectory at once—the one along which the action is minimized (more precisely, stationary). This unified formalism embraces classical mechanics, electrodynamics, field theory, and even quantum mechanics (through Feynman's path integrals).

Principle of Least Action

Action: $S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) , dt$, where $L = T - V$ is the Lagrangian of the system (kinetic energy minus potential energy). $q = (q_1, ..., q_n)$ are the generalized coordinates (e.g., angles, lengths), $\dot{q}$ are the generalized velocities.

Hamilton's Principle: The actual trajectory of the system between fixed states $q(t_1)$ and $q(t_2)$ is a stationary point of the action: $\delta S = 0$.

Lagrange Equations. Applying EL (Euler-Lagrange) to each $q_i$:

$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, \quad i = 1, ..., n. $

Here $\frac{\partial L}{\partial \dot{q}_i} = p_i$ is the generalized momentum, $\frac{\partial L}{\partial q_i}$ is the generalized force. The equation reads "the change in momentum equals the force"—this is an abstract generalization of Newton's second law.

Examples of Physical Systems

1. Simple Pendulum. The coordinate $\theta$ is the deviation angle. Kinetic energy $T = \frac{1}{2} m l^2 \dot{\theta}^2$, potential $V = mgl (1 - \cos \theta)$. Lagrangian $L = T - V$.

EL: $\frac{d}{dt} (m l^2 \dot{\theta}) - [-mgl \sin \theta] = 0 \rightarrow m l^2 \ddot{\theta} + m g l \sin \theta = 0 \rightarrow$ $\ddot{\theta} + \frac{g}{l} \sin \theta = 0$—the well-known pendulum equation. For small $\theta$: $\ddot{\theta} + \frac{g}{l}\theta = 0$—harmonic oscillator with period $T = 2\pi\sqrt{l/g}$.

Numerical example. For $l = 1$ m, $g = 9.81$ m/s², the period of small oscillations $T \approx 2.007$ s. For $\theta_0 = 60^\circ$ the exact (nonlinear) period increases by ~7% due to anharmonicity.

2. Particle in an Electromagnetic Field. $L = \frac{1}{2}mv^2 + \frac{q}{c} \mathbf{A} \mathbf{v} - q\phi$, where $\mathbf{A}$ is the vector potential, $\phi$ is the scalar potential, $q$ is the charge, $c$ is the speed of light. EL reproduces the Lorentz force $F = q\mathbf{E} + \frac{q}{c} \mathbf{v} \times \mathbf{B}$.

3. Fermat's Principle. Light propagates along the trajectory minimizing the optical time $T = \int \frac{n(\mathbf{r}) ds}{c}$. From the variational condition, Snell's law is derived: $n_1 \sin \theta_1 = n_2 \sin \theta_2$. A lens works because all points of refraction yield the same optical time from object to image.

Hamiltonian Mechanics

The Legendre transformation converts the Lagrangian $L$ into the Hamiltonian $H$:

$ p_i = \frac{\partial L}{\partial \dot{q}_i} \quad \text{(generalized momentum)}, $ $ H(q, p, t) = \sum_i p_i \dot{q}_i - L. $

Hamilton's equations: $\dot{q}_i = \frac{\partial H}{\partial p_i}$, $\dot{p}_i = -\frac{\partial H}{\partial q_i}$. This is a system of $2n$ first-order ODEs instead of $n$ second-order ODEs. For conservative systems $H = T + V$ is the total energy.

Advantage. In the Hamiltonian formulation, symmetries and conserved quantities are clearly visible. The phase space $(q, p)$ is the natural arena for statistical mechanics and chaos theory.

Noether’s Theorem

Noether’s Theorem (1918): Every continuous symmetry of the Lagrangian corresponds to a conservation law.

• Invariance in time $\rightarrow$ law of conservation of energy.
• Invariance under spatial translations $\rightarrow$ law of conservation of momentum.
• Invariance under rotation $\rightarrow$ law of conservation of angular momentum.

This fundamental correspondence underpins all modern physics, including the Standard Model of elementary particles.

Real Applications

• Robotics. The equations of motion for a multi-link manipulator (5–7 degrees of freedom) are derived via the Lagrangian—this is much simpler than spelling out the individual interaction forces. Inverse dynamics algorithms for KUKA and ABB robots are based on the Lagrangian formalism.
• Space Navigation. Calculations of gravitational maneuvers (Voyager, New Horizons) use Hamiltonian mechanics and perturbation theory. The Moon's orbit, Lagrange points L1–L5 (where the James Webb and SOHO telescopes are located) are consequences of the Hamiltonian structure of the three-body problem.
• Molecular Dynamics. Protein simulations (10⁵–10⁶ atoms) integrate the Hamiltonian equations with a step of ~1 fs. Symplectic integrators (Verlet) preserve energy over long times—critical for biophysics.
• Financial Physics (econophysics). Models of market dynamics use the Lagrangian formalism to describe the “motion” of prices under the influence of the “potentials” of supply and demand.

Assignment. Double pendulum: mass $m_1$ on rod $l_1$, mass $m_2$ on rod $l_2$ suspended from $m_1$. (a) Write out the kinetic and potential energies in coordinates $(\theta_1, \theta_2)$. (b) Write the Lagrangian $L$ and the Lagrange equations. (c) Explain why this system is chaotic (sensitivity to initial conditions, positive Lyapunov exponent). (d) Numerically (RK4, step 0.001 s, 30 s) solve for $l_1 = l_2 = 1$, $m_1 = m_2 = 1$, $\theta_1(0) = \theta_2(0) = \pi/4$ and for $\theta_2(0) = \pi/4 + 0.01$. Plot $\theta_1(t)$ for both cases on the same graph—visualize exponential divergence.