Stability, Equilibrium, and the Quadratic Lagrangian

The Geometry of Equilibrium

Imagine a ball resting in a valley. If you nudge it slightly, it rolls back toward the center. This is a stable equilibrium. Conversely, if the ball sits on a hilltop, a tiny nudge sends it rolling away. This is an unstable equilibrium. In analytical mechanics, we identify these states by looking at the potential energy surface. An equilibrium point is where the potential energy is stationary, meaning its first derivative with respect to the generalized coordinates is zero. However, the first derivative only tells us the slope is flat; it does not tell us if the flat spot is a bowl or a dome. To distinguish between the two, we must examine the curvature of the potential energy surface. This curvature is captured by the second derivatives, which form the Hessian matrix. If the curvature is positive in all directions, the system is stable. If it is negative in any direction, the system is unstable. This geometric intuition is the starting point for our mathematical treatment.

Expanding the Lagrangian

To analyze small oscillations, we expand the Lagrangian \( L = T - V \) about the equilibrium point \( q_0 \). Let \( q_i = q_{0i} + \eta_i \), where \( \eta_i \) is a small displacement. Since the first derivatives of the potential vanish at equilibrium, the potential energy \( V \) can be approximated by its Taylor series up to the second order:

We define the stiffness matrix \( K_{ij} \) as \( K_{ij} = \frac{\partial^2 V}{\partial q_i \partial q_j} \big|_{q_0} \). The kinetic energy \( T \) is quadratic in the velocities \( \dot{q}_i \). For small oscillations, we assume the mass matrix \( M_{ij} \) is constant and evaluated at equilibrium:

Thus, the quadratic Lagrangian is:

This form is crucial because it leads to linear equations of motion. The system behaves like a set of coupled springs and masses, where the coupling is determined by the off-diagonal elements of \( M \) and \( K \).

Equations of Motion and Stability

Applying the Euler-Lagrange equations to \( L_{\text{quad}} \) yields:

In matrix notation, this is written as:

To find the normal modes, we assume a solution of the form \( \boldsymbol{\eta}(t) = \mathbf{a} e^{i \omega t} \). Substituting this into the equation of motion gives:

For a non-trivial solution \( \mathbf{a} \neq 0 \), the determinant of the coefficient matrix must vanish:

This is the characteristic equation. The roots \( \omega_r^2 \) are the squared normal frequencies. If all \( \omega_r^2 \) are positive, the system is stable, and the motion is oscillatory. If any \( \omega_r^2 \) is negative, the corresponding mode is unstable, leading to exponential growth in displacement. The stability of the system is therefore determined by the eigenvalues of the generalized eigenproblem defined by the matrices \( K \) and \( M \).

Connection to Normal Modes

The solutions to the generalized eigenproblem are the normal modes of the system. Each mode \( r \) has a specific frequency \( \omega_r \) and a shape vector \( \mathbf{a}^{(r)} \). The general solution is a superposition of these modes:

Here, \( C_r \) and \( \phi_r \) are constants determined by the initial conditions. The normal modes are orthogonal with respect to the mass matrix, meaning \( \mathbf{a}^{(r)\top} M \mathbf{a}^{(s)} = 0 \) for \( r \neq s \). This orthogonality allows us to decouple the equations of motion into independent harmonic oscillators, each vibrating at its own natural frequency. This decoupling is the key insight that makes complex multi-degree-of-freedom systems tractable.