Theory, worked examples and 11 previous-year questions with full solutions — IIT JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards.
Theory
12.1 Equilibrium and the harmonic approximation
(a) Stable and unstable equilibrium
We consider a conservative system with \(n\) degrees of freedom whose potential energy \(V(q_1,\dots,q_n)\) depends on the generalized coordinates alone. We also assume that the transformation from Cartesian to generalized coordinates does not involve the time explicitly, so that time-dependent constraints are excluded and the kinetic energy is a homogeneous quadratic form in the generalized velocities (Section 10.3(c)). The system is in equilibrium at a configuration \((q_{01},\dots,q_{0n})\) when every generalized force \(-\partial V/\partial q_i\) vanishes there,
so that the potential energy is stationary at the equilibrium configuration. A system placed at such a configuration with zero velocities remains there indefinitely; a pendulum hanging at rest and a galvanometer coil at its zero position are familiar examples.
An equilibrium is stable if a small disturbance produces only a small, bounded motion about it. When \(V\) has a minimum at the equilibrium, stability follows from energy conservation. Suppose the system is given a small energy \(dE\) above the equilibrium value \(E_0\), as in Figure 12.1. Any departure from equilibrium raises \(V\), so the kinetic energy must fall, and the system can never move beyond the points where \(V=E_0+dE\); the motion is bounded. If, on the other hand, \(V\) decreases for some departure from equilibrium, the kinetic energy grows along that direction and the motion runs away: the equilibrium is unstable. The one-dimensional version of this argument, with energy diagrams and phase portraits, was given in Chapter 5.
Figure 12.1 A potential minimum at \(q_{0i}\). A system with energy \(E_0+dE\) is confined to a small neighbourhood of the minimum.
(b) The harmonic approximation
We are interested in motion close to a configuration of stable equilibrium. The deviations from equilibrium are denoted by
\[\xi_i=q_i-q_{0i},\]
and these may be used as new generalized coordinates. Since they are small, we expand the potential energy in a Taylor series about the equilibrium,
The constant term may be taken as the zero of potential energy, and the linear terms vanish because the equilibrium condition makes every first derivative zero. Terms of third and higher order in the \(\xi_i\) are negligible for sufficiently small displacements. Hence, to lowest order,
This is the harmonic approximation. For a system with one degree of freedom it reduces to \(V\approx\frac12V''(q_0)\,\xi^2\), and the coefficient \(V''(q_0)\) plays the part of the spring constant. If the kinetic energy is \(T=\frac12m\dot\xi^2\), the equation of motion \(m\ddot\xi=-V''(q_0)\xi\) describes simple harmonic motion with angular frequency
When the coordinate is not a Cartesian one, \(m\) is replaced by the coefficient \(M(q_0)\) in \(T=\frac12M(q)\dot q^2\), evaluated at equilibrium. If \(V''(q_0)<0\) the equilibrium is unstable and the deviation grows exponentially, as \(e^{\pm\kappa t}\) with \(\kappa^2=-V''(q_0)/m\). If \(V''(q_0)=0\) the harmonic approximation gives no restoring force at all, and the lowest non-vanishing term of the expansion governs the motion, which is then periodic but not simple harmonic (Section 12.1(c)).
Example 12.1 A particle of mass \(m\) moves along the positive \(x\)-axis in the potential \(V(x)=ax^2+\dfrac{b}{x^2}\), where \(a\) and \(b\) are positive constants. Find the angular frequency of small oscillations.
Example 12.2 A particle of mass \(m\) moves in the potential \(V(x)=\dfrac{V_0}{2}\left(e^{x}+e^{-x}\right)-V_0\), where \(V_0>0\) and \(x\) is measured in units in which it is dimensionless. Approximate the potential for small oscillations and find their angular frequency.
Solution. The potential is \(V=V_0(\cosh x-1)\), so \(V'(x)=V_0\sinh x\) vanishes at \(x=0\). Expanding the exponentials,
The Lagrangian for small oscillations is \(L=\frac12m\dot x^2-\frac12V_0x^2\), and Lagrange’s equation gives \(m\ddot x+V_0x=0\), so that
\[\omega=\sqrt{\frac{V_0}{m}}.\]
Had the potential been \(V_0-\frac12V_0(e^{x}+e^{-x})=-V_0(\cosh x-1)\approx-\frac12V_0x^2\), the point \(x=0\) would be a maximum. The equation of motion would then be \(m\ddot x-V_0x=0\), whose solutions grow as \(e^{\sqrt{V_0/m}\,t}\), and no small oscillation would exist. The sign of the second derivative must always be checked before a frequency is quoted.
(c) Oscillations beyond the harmonic approximation
Several examination questions concern bounded motion in potentials that are not quadratic near their minimum. For a particle of mass \(m\) in a symmetric potential well \(V(x)=V(-x)\) that increases monotonically for \(x>0\), energy conservation \(\frac12m\dot x^2+V(x)=E\) gives the time to travel from the origin to the turning point \(A\), where \(V(A)=E\). The full period is four times this:
For a power-law potential \(V(x)=k\lvert x\rvert^{n}\) with \(k,n>0\) the dependence on energy follows by scaling. The amplitude is \(A=(E/k)^{1/n}\). Substituting \(x=Au\) and \(V=Eu^n\),
For \(n=2\) the period is independent of the energy, which is the isochronism of the harmonic oscillator. For \(n=4\) we find \(T\propto E^{-1/4}\propto A^{-1}\), for \(n=6\) we find \(T\propto E^{-1/3}\), and for the linear potential \(n=1\) we find \(T\propto E^{1/2}\propto A^{1/2}\). The same results follow from the action variable of Chapter 11, but the scaling argument requires nothing beyond energy conservation. For the simple pendulum, \(V=mgl(1-\cos\theta)\) rises more slowly than the quadratic approximation \(\frac12mgl\theta^2\), so the restoring torque is weaker at large amplitude and the period increases with amplitude.
12.2 The matrix method
(a) The potential-energy matrix
For a system with several degrees of freedom the harmonic approximation of Section 12.1(b) gives
where \(\xi\) denotes the column of displacements \((\xi_1,\dots,\xi_n)^{\mathsf T}\). The constants \(V_{ij}\) form the potential-energy matrix\(\mathsf V\), which is real and symmetric because the order of differentiation does not matter. It is the generalization of the spring constant \(k\) of a single oscillator. In practice \(\mathsf V\) is read off from the quadratic form: a diagonal term \(\frac12c\,\xi_i^2\) contributes \(V_{ii}=c\), and a cross term \(c\,\xi_i\xi_j\)\((i\neq j)\) is split symmetrically as \(\frac12c\,\xi_i\xi_j+\frac12c\,\xi_j\xi_i\), giving \(V_{ij}=V_{ji}=c\). At a stable equilibrium \(V\) is a minimum, so \(\mathsf V\) is positive definite, or positive semi-definite when some displacements cost no energy.
(b) The kinetic-energy matrix
Because the coordinate transformation does not involve the time, the kinetic energy has the form \(T=\frac12\sum_{i,j}M_{ij}(q)\,\dot q_i\dot q_j\). Since \(\dot q_i=\dot\xi_i\), and the velocities are themselves small, the coefficients may be evaluated at the equilibrium configuration; the corrections are of third order and are consistently dropped. Hence
The kinetic-energy matrix\(\mathsf T\) is real, symmetric and positive definite, since the kinetic energy is positive for any non-zero motion. For particles moving along Cartesian axes it is simply the diagonal matrix of the masses.
Since \(\partial L/\partial\dot\xi_i=\sum_jT_{ij}\dot\xi_j\) and \(\partial L/\partial\xi_i=-\sum_jV_{ij}\xi_j\) (the symmetry of the matrices supplies the factor 2 that cancels the \(\frac12\)), Lagrange’s equations are
These are \(n\) coupled, linear, second-order equations with constant coefficients. The coupling may be through the potential (off-diagonal \(V_{ij}\), as for masses joined by springs) or through the kinetic energy (off-diagonal \(T_{ij}\), as for a double pendulum).
(d) The secular equation and the normal frequencies
We look for solutions in which every coordinate oscillates with the same frequency and phase,
\[\xi_j=C\,a_j\,e^{-i\omega t},\]
where the real numbers \(a_j\) fix the relative amplitudes and the complex constant \(C\) fixes the overall amplitude and phase; the physical motion is the real part. Substitution into the equations of motion gives
This is a set of \(n\) linear homogeneous equations for the \(a_j\). It has a non-trivial solution only if the determinant of the coefficients vanishes:
This is the secular equation, or characteristic equation, of the system. It is an algebraic equation of degree \(n\) in \(\omega^2\), whose roots \(\omega_1^2,\dots,\omega_n^2\) give the normal frequencies (eigenfrequencies) \(\omega_k\). The problem is a generalized eigenvalue problem \(\mathsf V\mathbf a=\omega^2\mathsf T\mathbf a\); when \(\mathsf T=m\mathsf 1\) it reduces to the ordinary eigenvalue problem for \(\mathsf V/m\).
The roots \(\omega_k^2\) are real, because \(\mathsf T\) and \(\mathsf V\) are real and symmetric and \(\mathsf T\) is positive definite. To see this, multiply \(\mathsf V\mathbf a_k=\omega_k^2\mathsf T\mathbf a_k\) on the left by \(\mathbf a_k^{*\mathsf T}\); the quadratic forms \(\mathbf a_k^{*\mathsf T}\mathsf V\mathbf a_k\) and \(\mathbf a_k^{*\mathsf T}\mathsf T\mathbf a_k\) are real, and the second is positive, so their ratio \(\omega_k^2\) is real. At a stable equilibrium the first form is also non-negative, so \(\omega_k^2\geq0\) and every \(\omega_k\) is real. A negative root would signal instability.
A root \(\omega_k=0\) corresponds to a displacement that does not change the potential energy. It is not an oscillation but a uniform translation or rotation of the whole system, and such zero modes occur whenever the system is free to move as a whole, as in a molecule that is not anchored. The lowest non-zero frequency is called the fundamental, and the higher ones are referred to as overtones. When two or more roots coincide, the frequency is said to be degenerate.
(e) Normal modes and their orthogonality
For each root \(\omega_k\) the equations \((\mathsf V-\omega_k^2\mathsf T)\mathbf a_k=0\) determine the column \(\mathbf a_k\) up to a constant factor. The pattern of motion in which every coordinate oscillates as \(\xi_j\propto a_{jk}\cos(\omega_kt+\phi_k)\), with the ratios \(a_{1k}:a_{2k}:\cdots\) fixed, is the \(k\)-th normal mode of the system. The factor left free is conventionally fixed by the normalization
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