Hamiltonian Mechanics

2 Conservation Laws

2.1 Noether theorem

In this section we go deeper into the appearance of conservation laws in the lagrangian formulation and, in particular, a beautiful and important theorem due to Emmy Noether relating conserved quantities to symmetries.

Later on, once we introduce hamiltonian mechanics, we will come back to the study of symmetries and conservation laws in a more sophisticated way.

We have already seen that, when the lagrangian is not explicitly time dependent, the total energy is a conserved quantity. Can we expect more?

Let’s make another example before entering into the gist of the matter.

There is no apparent obstacle that prevents us to expect other conserved quantities. However, before being able to give a precise statement we need to introduce some preliminary concepts.

Let \(M\) be a smooth manifold. A smooth map \(\phi : M \to M\) is called diffeomorphism if it is invertible and its inverse \(\phi ^{-1}\) is smooth. At each point \(q\in M\) we can define the differential of \(\phi \), i.e., the linear map \(\phi _*\) between tangent spaces (sometimes also denoted \(\dd \phi _q\))

\begin{equation} \phi _*: T_q M \to T_{\phi (q)} M, \quad \phi _* v := \phi _*(q) v := \dv {t} (\phi \circ \gamma ) \Big |_{t=0}, \quad v\in T_q M, \end{equation}

where \(\gamma :(-\epsilon , \epsilon )\subset \mathbb {R} \to M\) is a smooth curve such that \(\gamma (0)=q\), \(\gamma '(0) = v\). On a euclidean space, for instance, the simplest such curve is \(\gamma (t) = q + v t\).

In local coordinates, the matrix of the linear map \(\phi _*\) coincides with the Jacobian matrix

\begin{equation} \phi _*(q) = \dd \phi _q = \left (\frac {\partial \phi ^i(q)}{\partial q^j}\right )_{1\leq i,j\leq n}. \end{equation}

The differential can be used to push tangent vectors on \(M\) forward to tangent vectors on \(\phi (M)\), which is why it is also called pushforward.

A diffeomorphism \(\Phi : M \to M\) is called a symmetry of the mechanical system \((M,L)\) if \(L\) is invariant¹ with respect to \(\Phi \), i.e.²

\begin{equation} \label {eq:symmetry} \Phi ^* L (q,\dot q) := L\left (\Phi (q), \Phi _*\dot q\right ) = L(q,\dot q) \qquad \forall (q,\dot q) \in TM. \end{equation}

Note that in [Arn89, Chapter 4] terminology, the above definition would instead be that \((M,L)\) admits the mapping \(\Phi \).

A fundamental property of symmetries of mechanical systems is summarized by the following theorem.

The converse is generally not true: i.e., in general it is not true that if \(\Phi \) maps solutions to solutions, then \(\Phi \) is a symmetry.

We are almost there. Before being able to introduce Noether theorem we need one last definition.

A one–parameter family of diffeomorphisms

\begin{equation} \Phi _s : M \to M \end{equation}

is called one–parameter group of diffeomorphisms if the following hold:

• • \(\Phi _0 = \Id \);

• • \(\forall s, t \in \mathbb {R}, \quad \Phi _s(\Phi _t(q)) = \Phi _{s+t}(q)\quad \forall q\in M\).

These two properties give away the inverse of \(\Phi _s\): \((\Phi _s)^{-1} = \Phi _{-s}\).

As one can expect, if all \(\Phi _s\) in a one-parameter group of diffeomorphisms are also symmetries of some lagrangian \(L\) on a manifold \(M\), we call \(\Phi _s\) a one-parameter group of symmetries for \((M,L)\). We can finally state Noether³ theorem!

Although I prefer the presentation of the proof in [Arn89, Chapter 20.B], I will give here one with a slightly different perspective.

Let’s now look at more and less immediate consequences of our discovery, starting from some of the constants of motion that we have already encountered.

2.1.1 Homogeneity of space: total momentum

Consider a system of \(N\) particles with lagrangian (1.59),

\begin{equation} L = \frac 12 \sum _{k=1}^N m_k \dot {\vb *{x}}_k^2 - U(\vb *{x}_1, \ldots , \vb *{x}_N), \end{equation}

which is invariant with respect to translations

\begin{equation} \Phi _s(\vb *{x}_1, \ldots , \vb *{x}_N) := (\vb *{x}_1 + s \hat {\vb *{x}},\ldots , \vb *{x}_N + s \hat {\vb *{x}}), \end{equation}

for any direction \(\hat {\vb *{x}}\), \(|\hat {\vb *{x}}|=1\) and all \(s\in \mathbb {R}\). I.e.⁵

\begin{equation} L(\vb *{x}_1, \ldots , \vb *{x}_N, \dot {\vb *{x}}) = L(\vb *{x}_1 + s \hat {\vb *{x}}, \ldots , \vb *{x}_N + s \hat {\vb *{x}}, \dot {\vb *{x}}), \end{equation}

which is the statement that space is homogeneous and a translation of the system in some direction \(\hat {\vb *{x}}\) does nothing to the equations of motion. These translations are elements of the Galilean group that we met in Chapter 1.2.

We call total momentum the vectorial quantity

\begin{equation} \vb *{P} = \sum _{k=1}^N m_k \dot {\vb *{x}}_k = \sum _{k=1}^N \vb *{p}_k. \end{equation}

As we saw in Example 2.2, Noether theorem implies that for any \(\hat {\vb *{x}}\), the projection of the total impulse on the direction \(\hat {\vb *{x}}\), which is

\begin{equation} \langle \vb *{P}, \hat {\vb *{x}}\rangle = \sum _{k=1}^N \langle \vb *{p}_k, \hat {\vb *{x}}\rangle , \end{equation}

is a conserved quantity. Since \(\hat {\vb *{x}}\) is arbitrary, this means that also \(\vb *{P}\) is conserved.

This should be intuitively clear: one point in space is much the same as any other. So why would a system of particles speed up to get somewhere else, when its current position is just as good? This manifests itself as conservation of momentum.

This fact can be also interpreted in the following way: the sum of all forces acting on the point particles in the closed system is zero, i.e., Newton’s third law \(\sum _k \vb *{F}_k = 0\). Which, you have already figured out by solving Exercise 1.1.

2.1.2 Isotropy of space: angular momentum

The isotropy of space is the statement that a closed system is invariant under rotations around a fixed axis. To work out the corresponding conserved quantities one can work directly with the infinitesimal form of the rotations.

Let’s first look at the invariants related to isotropy: rotations around an axis in \(\mathbb {R}^3\) are identifiable with orthogonal transformation \(A\in SO(3)\), the special orthogonal group – orthogonal matrices \(A^TA = \Id \) with determinant \(1\), where we are denoting by \(\Id \) the identity matrix.

For future reference, it is worth recalling in more generality the structure of these transformations. We will do it briefly, for more details on the special orthogonal groups and their relation to rotations and vector products, you can refer to [MR99].

Consider, for some small \(\epsilon > 0\), a family of orthogonal matrices around the identity

\begin{equation} \label {eq:familyson} A(s) \in SO(n), \quad A(0) = \Id , \quad |s| < \epsilon . \end{equation}

If we look at \(SO(n)\) as the smooth manifold in the space \(M^{n\times n}\) of \(n\times n\) matrices defined by the equations

\begin{equation} A^T A = \Id ,\quad \det A = 1, \end{equation}

then its tangent space over the identity is

\begin{equation} T_{\Id } SO(n) = \big \{X\in M^{n\times n}\mid X^T = -X\big \}, \end{equation}

the space of antisymmetric \(n\times n\) matrices.

This is a consequence of the following proposition.

In view of Exercise 2.4, Noether theorem tells us that the conserved quantity for a single particle system would be

\begin{equation} I = \langle \vb *{p}, X\vb *{x}\rangle = \langle \vb *{p}, \vb *{N}\wedge \vb *{x}\rangle = \langle \vb *{N}, \vb *{x}\wedge \vb *{p} \rangle , \end{equation}

where for the second equality we used the relation between \(3\times 3\) skew-symmetric matrices and vector products, and for the last we used the cyclicity of the vector product. The quantity

\begin{equation} \vb *{M} = \vb *{x} \wedge \vb *{p} \end{equation}

is called angular momentum.

\begin{equation} \vb *{M} = \sum _{k=1}^N \vb *{x}_k \wedge \vb *{p}_k \end{equation}

are conserved quantities.

2.1.3 Homogeneity of time: the energy

We discussed homogeneity of space and isotropy, and discovered that they are related to the conservation of different types of momentum. One could now ask, what about homogeneity of time?

Homogeneity of time means that \(L\) is invariant with respect to the transformation \(t\mapsto t+s\), which in turn means \(\frac {\partial L}{\partial t} = 0\). We already saw in Section 1.3.2 that for natural lagrangians this corresponds to conservation of energy! In other words, Time is to Energy as Space is to Momentum.

If you have taken a course on special relativity, you have probably seen that the combination of energy and \(3\)-momentum forms a \(4\)-vector which rotates under space-time transformations. You don’t have to be Einstein to make the connection: the link between energy-momentum and time-space exists also in Newtonian mechanics.

2.1.4 Scale invariance: Kepler’s third law

We have mentioned in Remark 1.4 that the two lagrangians \(L\) and \(\alpha L\), \(\alpha \neq 0\), give rise to the same equations of motion. This does not look like a symmetry in our sense, but it is worth investigating its meaning in light of what we have seen so far.

2.2 The spherical pendulum

Let’s come back to the spherical pendulum, introduced in Example 1.13. Assume it moves in the presence of constant gravitational acceleration \(g\) pointing downwards. It’s an exercise in trigonometry to see that the lagrangian in spherical coordinates takes the form

\begin{equation} L = \frac m2 l^2 \left (\dot \theta ^2 + \dot \phi ^2 \sin ^2\theta \right ) + mgl \cos \theta . \end{equation}

We can immediately see that \(\phi \) is a cyclic coordinate, as such the corresponding kinetic momentum

\begin{equation} \label {eq:spI} I = \frac {\partial L}{\partial \dot \phi } = m l^2 \dot \phi \sin ^2 \theta \end{equation}

is a constant of motion. With our choice coordinates, \(I\) corresponds to the component of the angular momentum in the \(\phi \) direction. We can now use (2.65) to replace all the appearances of \(\dot \phi \) with

\begin{equation} \dot \phi = \frac {I}{ml^2 \sin ^2\theta }. \end{equation}

Computing the remaining equation of motion for \(\theta \), we get

\begin{equation} ml^2 \ddot \theta = ml^2 \dot \phi ^2\sin \theta \cos \theta - mgl \sin \theta = -ml^2\frac {\partial V_{\mathrm {eff}}}{\partial \theta } \end{equation}

where

\begin{equation} V_{\mathrm {eff}} = -\frac {g}l\cos \theta + \frac {I^2}{2m^2l^4}\frac 1{\sin ^2\theta }. \end{equation}

We are left with one-degree of freedom in \(\theta \), thus we can integrate by quadrature using the energy. Indeed, the conservation of energy translates into

\begin{equation} E = \frac {\dot \theta ^2}2 + V_{\mathrm {eff}}(\theta ), \end{equation}

where \(E>\min _\theta V_{\mathrm {eff}}(\theta )\) is a constant, and therefore

\begin{equation} \label {eq:sptheta} t - t_0 = \frac 1{\sqrt 2}\int \frac {\dd \theta }{\sqrt {E - V_{\mathrm {eff}}(\theta )}} \end{equation}

Thanks to (2.65), we can use (2.70) to solve for \(\phi \):

\begin{equation} \phi = \int \frac {I}{ml^2} \frac 1{\sin ^2\theta } \;\dd t = \frac {I}{\sqrt {2} m l^2} \int \frac 1{\sin ^2\theta } \frac {1}{\sqrt {E - V_{\mathrm {eff}}(\theta )}}\; \dd \theta \end{equation}

Close to a minimum of the effective potential \(V_{\mathrm {eff}}\), we have a bounded oscillatory motion in \(\theta \) while the constant sign of \(\dot \phi \) implies that the particle keeps spinning with various degrees of acceleration in the \(\phi \) direction.

At a minimum \(\theta _0\) of the effective potential, which occurs by a careful balancing of the energy and the angular momentum, we have a stable orbit. Looking at small oscillations around this point as we did for the simple pendulum, one can prove that small oscillations about \(\theta = \theta _0\) have a frequency \(\omega ^2 = \frac {\partial ^2 V_{\mathrm {eff}}(\theta _0)}{\partial \theta ^2}\).

As you can see, the existence of two integral of motion allowed us to practically solve the equations of motion and describe many details of the behavior of the system.

This is a far more general statement that we will see again at work for the Kepler’s problem and for motions in central potentials, and that, once we introduce hamiltonian systems, will be at the core of integrable systems and perturbation theory.

2.3 Intermezzo: small oscillations

We have already seen a few cases in which, by a small perturbation near a stable equilibrium, we could describe with various degrees of accuracy the behavior of mechanical systems in terms of the motion of an harmonic oscillator.

In this section we will see that this phenomenon is much more general (and useful) by showing that the motion of systems close to equilibrium is described by a number of decoupled simple harmonic oscillators, each ringing at a different frequency.

Let’s revisit what we saw in Sections 1.3.1 and 1.3.3 in slightly more generality and in a more organic way. Let’s consider the Newton equation for a closed system with one degree of freedom in the presence of conservative forces, i.e., an equation like (1.72):

\begin{equation} \ddot x = F(x). \end{equation}

As we saw in Example 1.10, it makes sense to ask ourselves what does the motion look like around an equilibrium point \(x^*\), i.e., a point such that \(F(x^*) = 0\). To analyze this in more details, we linearize the equation. Let \(x(t)\) be a small perturbation around \(x^*\). For convenience,

\begin{equation} x(t) = x^* + \gamma (t), \end{equation}

where \(\gamma \) is small so that we can expand \(F\) in Taylor series around \(x^*\), getting

\begin{equation} \ddot \gamma = \frac {\dd F}{\dd x}(x^*)\gamma + O(\gamma ^2). \end{equation}

Neglecting the higher order terms, we can see that there are three possibilities.

• • If \(\frac {\dd F}{\dd x}(x^*) < 0\), the system behaves like a spring (1.5) with \(\omega _0^2 = -\frac {\dd F}{\dd x}(x^*)\). The general solution (1.6) tells us that the system undergoes stable oscillations around \(x^*\) with frequency \(\omega _0\) (and period \(\tau _0 = 2\pi /\omega _0\)).

• • If, on the other hand, \(\frac {\dd F}{\dd x}(x^*) > 0\), the force is pushing the system away from the equilibrium: the solution is of the form \(\gamma (t) = C_+ e^{\lambda t} + C_- e^{-\lambda t}\), for some integration constants \(C_+\) and \(C_-\) and with \(\lambda ^2 = \frac {\dd F}{\dd x}(x^*)\). There are now two possibilities: either \(C_+ = 0\) and \(x(t) \to x^*\) as \(t\) grows, or \(x\) grows exponentially and, in a short time, leaves the region of good approximation. In this last case the system is said to be linearly unstable.

• • Finally, there is the case \(\frac {\dd F}{\dd x}(x^*) = 0\), corresponding to a neutral equilibrium: the solution is a uniform motion of the form \(\gamma (t) = C_0 + C_1 t\). We will come back to this case in a couple of sections.

If we generalize the discussion above to \(n\) degrees of freedom, and we consider a natural lagrangian

\begin{equation} L(q,\dot q) = \frac 12 g_{ij} \dot q^i \dot q^j - U(q^1, \ldots , q^n), \end{equation}

the equilibrium condition translates to the fact that \(q_* = (q_*^1, \ldots , q_*^n)\) must satisfy

\begin{equation} \frac {\partial U}{\partial q^k} (q_*) = 0 \quad k=1,\ldots ,n. \end{equation}

To study its stability we now need to consider the hessian of the potential around \(q_*\):

\begin{equation} h = (h_{ij}),\quad h_{ij} = \frac {\partial ^2 U}{\partial q^i\partial q^j} (q_*). \end{equation}

Neglecting the higher order terms in the Taylor expansion of the polynomial, the linearized system near to the equilibrium point is given by the lagrangian [Arn89, Chapter 22.D]

\begin{equation} \label {eq:genlagso} L_0 = \frac 12 g_{ij} \dot q^i \dot q^j - \frac 12 h_{ij} q^i q^j. \end{equation}

Motions in a linearized system \(L_0\) near an equilibrium point \(q =q_*\) are called small oscillations. In a one-dimensional problem the numbers \(\tau _0\) and \(\omega _0\) are called period and frequency of the small oscillations.

Unfortunately the Euler-Lagrange equations look horrible compared to the one dimensional case... unless we find a way to simultaneously diagonalize \(g\) and \(h\).

As you may have guessed, linear algebra comes to the rescue: given two symmetric matrices \(h,g\), such that \(g\) is positive definite, there exists an invertible matrix \(\Gamma \) such that

\begin{equation} \Gamma ^T g \Gamma = I, \quad \Gamma ^T h \Gamma = \diag (\lambda _1, \ldots , \lambda _n), \end{equation}

where \(\lambda _1, \ldots , \lambda _n\) are the roots of the characteristic polynomial [Arn89, Chapter 23.A]

\begin{equation} \label {eq:charso} \det (h - \lambda g) = 0. \end{equation}

In the new coordinates \(\widetilde q = \Gamma q\), the lagrangian \(L_0\) takes the form

\begin{equation} L_0(\widetilde q, \dot {\widetilde q}) = \frac 12 \sum _{k=1}^n \left (\|\dot {\widetilde q}^k\|^2 - \lambda _k (\widetilde q^k)^2 \right ), \end{equation}

whose equations of motion read

\begin{equation} \ddot {\widetilde q}^k = -\lambda _k \widetilde q^k. \end{equation}

We have proven the following theorem.

In view of the observations above, the solution \(\widetilde q(t) = q^*\) is stable if and only if all the eigenvalues \(\lambda _1, \ldots , \lambda _n\) are positive:

\begin{equation} \lambda _i = \omega _i^2, \quad \omega _i > 0. \end{equation}

In fact, by the nature of \(\Gamma \), the coordinate system \(\widetilde q\) is orthogonal with respect to the scalar product \((g\dot q, \dot q)_{\mathbb {R}^n}\) and therefore the theorem above can be restated as

Which immediately implies the following.

As well exemplified from Example 1.12, a sum of characteristic oscillations is generally not periodic!

2.3.1 Decomposition into characteristic oscillations

In view of what we have seen, we can write the characteristic oscillations in the form \(q(t) = e^{i\omega t} \vb *{\xi }\), then by substituting in the Euler-Lagrange equations,

\begin{equation} \dv {t} g \dot q = h q, \end{equation}

we find

\begin{equation} (h - \omega ^2 g)\vb *{\xi } = 0. \end{equation}

Solving (2.80), we find the \(n\) eigenvalues \(\lambda _k = \omega _k^2\), \(k=1, \ldots , n\), to which we can associate \(n\) mutually-orthogonal eigenvectors \(\vb *{\xi }_k\), \(k=1, \ldots , n\).

If \(\lambda \neq 0\) a general solution then has the form

\begin{equation} q(t) = \mathbb {R}e \sum _{k=1}^n C_k e^{i \omega _k t}\vb *{\xi }_k, \end{equation}

even in the case some eigenvalues come with multiplicities.

In conclusion, to decompose a motion into characteristic oscillations, it’s enough to project the initial conditions onto the characteristic directions, i.e., the eigenvectors of \(h - \lambda g\), and solve the corresponding one-dimensional problems.

More on small oscillations and their applications, with plenty of examples, can be found in [Arn89, Chapters 23–25].

2.3.2 The double pendulum

Let’s revisit the double pendulum, already introduced in Example 1.11. For simplicity, let’s assume that the masses are the same, \(m = m_1 = m_2\), as well as the two lengths \(l = l_1 = l_2\). The full lagrangian then reads

\begin{equation} L = m l^2 \dot x_1^2 + \frac m2 l^2 \dot x_2^2 + m l^2 \cos (x_1 -x_2)\dot x_1 \dot x_2 + 2ml g \cos x_1 + mlg\cos x_2. \end{equation}

One can check that the stable equilibrium point is \(x_1 = x_2 = 0\). Linearizing the lagrangian around this point and dropping the (irrelevant) constant terms, we obtain:

\begin{equation} L_0 = m l^2 \dot x_1^2 + \frac m2 l^2 \dot x_2^2 + m l^2 \dot x_1 \dot x_2 - ml g x_1^2 - \frac 12 mlg x_2^2. \end{equation}

The corresponding Euler-Lagrange equations are then given by

\begin{align} & 2ml^2 \ddot x_1 + ml^2 \ddot x_2 = -2mgl x_1 \\ & ml^2 \ddot x_1 + ml^2 \ddot x_2 = -mgl x_2, \end{align} which in matrix form becomes

\begin{equation} \begin{pmatrix}2&1\\1&1\end {pmatrix} \ddot {\vb *{x}} = - \frac gl \begin{pmatrix}2&0\\0&1\end {pmatrix} \vb *{x}. \end{equation}

Inverting the left hand side of the equations of motion to end up with

\begin{equation} \label {eq:doublep} \ddot {\vb *{x}} = - \frac gl \begin{pmatrix}2&-1\\-2&2\end {pmatrix} \vb *{x}. \end{equation}

Diagonalizing the matrix on the right hand side, is equivalent to perform the transformation shown in Section 2.3.1.

The matrix in (2.92) has eigenvalues \(\lambda _\mp = \omega _\mp ^2 = \frac {g}l \left (2\mp \sqrt {2}\right )\), the square of the normal frequencies, with corresponding (non normalized) eigenvectors, the normal modes, \(\vb *{\xi }_\mp = \left (1, \pm \frac 1{\sqrt {2}}\right )^T\). The corresponding motions are then given by

\begin{equation} \vb *{x}(t) = C_+ \cos (\omega _+ (t - t_0)) \vb *{\xi }_+ + C_- \cos (\omega _- (t - t_0)) \vb *{\xi }_-. \end{equation}

Note that the oscillation frequency for the mode in which the rods oscillate in the same direction, \(\vb *{\xi }_-\), is lower than the frequency of the mode in which they are out of phase, \(\vb *{\xi }_+\).

2.3.3 The linear triatomic molecule

Let’s consider the example of a linear triatomic molecule (e.g., carbon dioxide \(CO_2\)). This consists of a central atom of mass \(M\) flanked symmetrically by two identical atoms of mass \(m\). For simplicity, let’s assume that the atoms are aligned and the motion of each atom happens only in the direction parallel to the molecule.

If \(x_i\), \(i=1,2,3\), denote the displacement of the atoms in the molecule, the lagrangian for such object takes the form,

\begin{equation} L = \frac {m\dot x_1^2}2 + \frac {M\dot x_2^2}2 + \frac {m\dot x_3^2}2 - V(x_1 - x_2) - V(x_2 - x_3). \end{equation}

The function \(V\) is, in general, a complicated interatomic potential, however this will not matter for us, as we are interested here in the oscillations around the equilibrium. Indeed, assume \(x_i = x_i^*\) is an equilibrium position. By symmetry, we have \(|x_1^*-x_2^*| = |x_2^* - x_3^*| = r_*\). Taylor expanding the potential around the equilibrium point and dropping the \(O\left (|r-r_*|^3\right )\) terms, we get

\begin{equation} V(r) = V(r_*) + \frac {\partial V}{\partial r}\Big |_{r=r_*}(r-r_*) + \frac 12 \frac {\partial ^2 V}{\partial r^2}\Big |_{r=r_*}(r-r_*)^2. \end{equation}

The first term is a constant and can be ignored, while the second term – being \(r_*\) an equilibrium – vanishes. Calling \(k = \frac {\partial ^2 V}{\partial r^2}\big |_{r=r_*}\) and \(q_i = x_i - x_i^*\), we end up with the lagrangian

\begin{equation} L = \frac {m \dot q_1}{2} + \frac {M \dot q_2}{2} + \frac {m\dot q_3^2}2 - \frac k2 \left ((q_1 - q_2)^2 + (q_2 - q_3)^2\right ), \end{equation}

where as usual we omitted the higher order corrections.

Recalling our notation, we are left with

\begin{equation} g = \begin{pmatrix}m&0&0\\0&M&0\\0&0&m\end {pmatrix}, \quad h = k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end {pmatrix}. \end{equation}

The normal modes and their frequencies are then obtained from the pairs of eigenvalues and eigenvectors of \(h - \omega ^2 g\). In this case, they can be easily computed as:

• • \(\lambda _1 = 0\), \(\vb *{\xi }_1 = (1,1,1)^T\): this is not an oscillation, instead it corresponds to a translation of the molecule.

• • \(\lambda _2 = \frac km\), \(\vb *{\xi }_2 = (1,0,-1)^T\): the external atoms oscillate out of phase with frequency \(\omega _2 = \sqrt {\frac km}\), while the central one remains stationary.

• • \(\lambda _3 = \frac km \left (1+2 \frac mM\right )\), \(\vb *{\xi }_3 = \left (1,-2\frac mM,1\right )^T\). This oscillation is more involved: the external atoms oscillate in the same direction, while the central atom moves in the opposite direction.

The corresponding small oscillations are given, in general, as the linear combinations of the normal modes above:

\begin{equation} \vb *{x}(t) = \vb *{x}^* + (C_1 + \widetilde C_1 t) \vb *{\xi }_1 + C_2 \cos (\omega _2 (t-t_2)) \vb *{\xi }_2 + C_3 \cos (\omega _3 (t-t_3)) \vb *{\xi }_3. \end{equation}

2.3.4 Zero modes

Recall Noether theorem (Theorem 2.2): every continuous one–parameter group of symmetries \(\Phi _s = Q(s,q)\) is associated to an integral of motion \(I = p_i \frac {\partial Q^i}{\partial s}\big |_{s=0}\).

Assume that we have \(L\) one–parameter families \(\Phi ^l_s = Q_l(s,q)\), \(l=1,\ldots ,L\). For small oscillations near an equilibrium \(q_*\), let’s write \(q^k = q^k_* + \vb *{\eta }^k\). Then the \(L\) first integrals can be rewritten as

\begin{equation} I_l = C_{li} \dot \eta ^i,\quad l=1,\ldots , L, \qquad \mbox {where}\qquad C_{li} = g_{ij} \frac {\partial Q^j_l}{\partial s}\Big |_{s=0}. \end{equation}

Therefore, we can define the (non normalized) normal modes

\begin{equation} \vb *{\xi }_l = C_{li} \vb *{\eta }^i, \quad l=1,\ldots , L, \end{equation}

which satisfy

\begin{equation} \ddot {\vb *{\xi }}_l = 0, \quad l=1,\ldots , L. \end{equation}

Thus, in systems with continuous symmetries, to each such continuous symmetry there is an associated zero mode, i.e., a mode with \(\omega _l = 0\), of the small oscillations problem.

We will revisit small oscillations once we discuss perturbation theory in the context of Hamiltonian systems. Before introducing the necessary formalism and discussing integrability and perturbation theory, we will spend some time looking at an important family of examples in light of our newly acquired knowledge.

2.4 Motion in a central potential

In the remainder of this section, we will study the symmetries of systems of the form (2.107) to study its dynamics.

We know from Section 2.1.2, that angular momentum is a first integral of rotationally symmetric lagrangians. This means that the lagrangian (2.107) has three constants of motion given by the three components of the angular momentum

\begin{equation} \vb *{M} = \vb *{x}\wedge \vb *{p}. \end{equation}

For convenience, let’s rewrite the equations of motion for (2.107) as

\begin{equation} m\ddot {\vb *{x}} = - U'(x)\frac {\vb *{x}}{x}, \quad x = \|\vb *{x}\|, \quad U'(x)=\frac {\dd U(x)}{\dd x}. \end{equation}

Introducing polar coordinates \((r,\phi )\) in the plane (2.110), our lagrangian transforms to

\begin{equation} L = \frac {m}{2} \left (\dot r^2 + r^2 \dot \phi ^2\right ) - U(r), \end{equation}

i.e., a system of two-degrees of freedom with a cyclic coordinate \(\phi \). If we denote \(z\) the component parallel to the angular momentum, the first integral corresponding to \(\phi \) is

\begin{equation} \label {eq:cyclicphi} \frac {\partial L}{\partial \dot \phi } = m r^2 \dot \phi = M_z. \end{equation}

The final form of the energy (2.115) has a deep meaning. It is the same as the energy of a closed mechanical systems with one degree of freedom under the influence of the effective potential

\begin{equation} \label {eq:effpotcp} U_{\mathrm {eff}}(r) := U(r) + \frac {M^2}{2 m r^2}. \end{equation}

In other words, we reduced the description of the radial motion to a one-dimensional closed system with natural lagrangian

\begin{equation} \label {eq:efflagcp} L = \frac {m \dot r^2}{2} - U_{\mathrm {eff}}(r). \end{equation}

Given two consecutive roots \(r_m\), \(r_M\) of the equation \(U_{\mathrm {eff}}(r) = E\), the radial motion is bounded to stay in the potential well \(r_m \leq r \leq r_M\). In the plane, this translates into being confined into an annular region as depicted in Figure 2.1.

2.4.1 Kepler’s problem

Let’s go back to Example 2.7, and replace \(U\) with the Newtonian gravitational potential \(k/\|\vb *{x}\|\) (see Examples 1.2 and 1.6). The effective potential (2.120), in this case is

\begin{align} U_{\mathrm {eff}} (r) = -\frac kr + \frac {M^2}{2 m r^2}, \quad & k = G \frac {m_1^2 m_2^2}{m_1+m_2} > 0, \\ & m = \frac {m_1m_2}{m_1+m_2}, \quad M := M_z. \end{align} It’s an exercise in calculus now, to see what this potential looks like. The effective potential has a vertical asymptote at \(r=0\), where it blows up to \(+\infty \), and the horizontal asymptote \(y=0\) for \(r\to +\infty \). Finally, the potential presents a critical point, corresponding to a global minimum,

\begin{equation} U_{\mathrm {eff}}^* := U_{\mathrm {eff}}\left (\frac {M^2}{k m}\right ) = - \frac {k^2m}{2M^2}, \end{equation}

after which the potential grows approaching \(0\) as \(r\to +\infty \).

In terms of phase portrait, this means that the motion is bounded if and only if

\begin{equation} - \frac {k^2 m}{2 M^2} < E < 0, \end{equation}

and it is unbounded for \(E \geq 0\).

The shape of the trajectory is then given by (2.119), which is explicitly solvable in this case:

\begin{equation} \phi = \arccos \left (\frac {\frac {M}{r}-\frac {km}{M}}{\sqrt {2mE + \frac {k^2m^2}{M^2}}}\right ) + C, \end{equation}

where the constant depends on the initial conditions. Choosing \(C=0\), we can rewrite the equation of the trajectory as

\begin{equation} \label {eq:KeplerConic} r = \frac {p}{1+e\,\cos \phi }, \qquad p:=\frac {M^2}{km}, \quad e:=\sqrt {1+\frac {2EM^2}{k^2 m}}, \end{equation}

which is the equation of a conic section with a focus at the origin, focal parameter \(p\) and eccentricity \(e\), in agreement with Kepler’s first law. Choosing \(C=0\), means that we have chosen the angle \(\phi \) such that \(\phi =0\) corresponds to the perihelion of the trajectory, i.e., the closest point to the focus.

We can now confirm the last claim in Example 1.2. Indeed, with some knowledge about conic sections one can conclude that (2.129) is an ellipse for \(E<0\), when \(0<e<1\). The semiaxes of the ellipse are then given by

\begin{equation} a = \frac {p}{1-e^2}, \quad b = \frac {p}{\sqrt {1-e^2}}. \end{equation}

Furthermore, when \(E=0\), i.e., \(e = 1\), we have a parabola and, finally, for \(E>0\), i.e., \(e >1\) an hyperbola.

We can also look at the time one body spends along parts of the orbit by looking at (2.117):

\begin{equation} t = \sqrt {\frac {m}{2}} \int \frac {\dd r}{\sqrt {E - U_{\mathrm {eff}}(r)}}, \quad U_{\mathrm {eff}}(r) = -\frac kr + \frac {M^2}{2 m r^2}. \end{equation}

In the case of elliptic trajectories \(0<e<1\), despite its look, we can actually solve it with some smart chains of substitutions. Using

\begin{equation} M = \sqrt {kmp}, \quad E = -k \frac {1-e^2}{2p}, \quad \mbox {and}\quad p= a(1-e^2), \end{equation}

and then the substitution

\begin{equation} \label {eq:rkp} r - a = - a e \cos \theta , \end{equation}

the integral becomes

\begin{align} t & = \sqrt {\frac {ma}k} \int \frac {r\dd r}{\sqrt {-r^2 + 2ar -ap}} \\ & = \sqrt {\frac {ma}k} \int \frac {r\dd r}{\sqrt {a^2e^2 - (r-a)^2}} \\ & = \sqrt {\frac {ma^3}{k}} \int (1- e \cos \theta ) \dd \theta \\ & = \sqrt {\frac {ma^3}{k}} (\theta - e \sin \theta ) + C. \label {eq:tkp} \end{align} By the arbitrariness of the choice of \(t_0\), we can always choose \(C=0\).

Together, (2.133) and (2.137), provide a parametrization of the radial motion:

\begin{equation} \left \lbrace \begin{aligned} r & = a(1-e\cos \theta ) \\ t & = \sqrt {\frac {ma^3}k}(\theta - e\sin \theta ) \end {aligned} \right .. \end{equation}

Given that \(\theta \mapsto r(\theta )\) is \(2\pi \) periodic, we can immediately obtain the period from \(t(\theta )\):

\begin{equation} T = 2\pi \sqrt {\frac {ma^3}k}. \end{equation}

To rewrite the orbits in cartesian coordinates, \(x = r \cos \phi \), \(y=r \sin \phi \), we can use the two representations for \(r\)

\begin{equation} \frac {p}{1+e\cos \phi } = r = a(1-e\cos \theta ) \end{equation}

to derive

\begin{equation} \cos \phi = -\frac {e - \cos \theta }{1-e \cos \theta },\quad \sin \phi = \frac {\sqrt {1-e^2}\;\sin \theta }{1-e\cos \theta }, \end{equation}

which, in turn, imply the parametrization in cartesian coordinates

\begin{equation} \left \lbrace \begin{aligned} x & = a(\cos \theta - e) \\ y & = a\sqrt {1-e^2}\;\sin \theta \\ t & = \sqrt {\frac {ma^3}k}(\theta - e\sin \theta ) \end {aligned} \right .. \end{equation}

It is possible to obtain an explicit dependence on time with some extra effort by expanding the trigonometric terms in terms of Bessel functions. This is rather useful when studying certain type of orbit-orbit or spin-orbit resonances, but we will not investigate it further in this course. For additional information you can refer to [AKN06; Arn10].

In Example 1.2, we mentioned a third conserved quantity for the Kepler problem.

2.5 D’Alembert principle and systems with constraints

In most of the examples above, we have started by assuming that each particle is free to roam in \(\mathbb {R}^3\). However, in many instances this is not the case. In this section we will briefly investigate how Lagrangian mechanics deals with these cases. For a different take on this problem, refer to [Arn89, Chapter 21].

If a free point particle living on a surface \(M\) in \(\mathbb {R}^3\) was evolving according to the free euclidean Newton’s law (without any potential), the particle would generally move in a straight line and not stay confined on the surface. From the point of view of Newton, this means that there should be a virtual force that keeps the particle constrained to the surface. Of course, constraints could be more general than this.

Let’s try again. Consider now a natural lagrangian (1.59)

\begin{equation} L(\vb *{x}, \dot {\vb *{x}}) = \sum _{k=1}^N \frac {m_k \|\dot {\vb *{x}}_k\|^2}{2} - U(\vb *{x}_1, \ldots , \vb *{x}_N) \end{equation}

whose point particles are forced to satisfy a number of holonomic constraints, i.e., relationships between the coordinates of the form

\begin{equation} \label {eq:holonomic} f_1(\vb *{x}_1, \ldots ,\vb *{x}_N) = 0, \;\ldots ,\; f_l(\vb *{x}_1, \ldots ,\vb *{x}_N) = 0 \end{equation}

where \(f_i : \mathbb {R}^{3N}\to \mathbb {R}\) smooth, \(i=1,\ldots ,l\).

Hamilton’s principle is then applied to curves which satisfy the relations (2.152), and whose fixed end points satisfy (2.152).

To be more precise, assume that the \(l\) equations (2.152) are independent, i.e., that they define a smooth submanifold \(Q\subset M\) of the configuration space \(M=\mathbb {R}^{3N}\) with maximal dimension \(\dim Q = 3N-l\). The tangent space \(T_qQ \subset \mathbb {R}^{3N}\) is a linear subspace of the ambient tangent space \(T_q \mathbb {R}^{3N}\) and its orthogonal complement is a linear subspace of dimension \(l\) spanned by the gradients of the functions \(f_1, \ldots , f_l\). For a brief but more comprehensive discussion of this you can look at [Ser20, Chapter 2.8].

Equations (2.153) and (2.158) fully determine the evolution of the constrained system and the constraint forces.

Constrained motions have also a nice intrinsic description in terms of a mechanical system on \(TQ\). Indeed, introducing local coordinates \(q= (q_1, \ldots , q_n)\in Q\), where \(n = 3N-l\), the substitution \(\vb *{x}_k = \vb *{x}_k(q)\), \(k=1,\ldots ,N\) immediately implies the following theorem.

This can be further generalized to mechanical systems which are constrained on the tangent bundle \(TM\) of a manifold \(M\) of dimension \(m\). In such case, let \(L=L(x,\dot x)\) be a non degenerate lagrangian on \(TM\), and consider a smooth submanifold \(Q\subset M\) of dimension \(n\) with local coordinates \((q^1,\ldots ,q^n)\). Assume that \(Q\) is regular with respect to the lagrangian, i.e., such that the restriction of \(\left (\frac {\partial ^2 L}{\partial \dot x^i \partial \dot x^j}\right )_{i,j=1,\ldots ,m}\) to \(Q\) is still non degenerate:

\begin{equation} \det \left (\frac {\partial ^2 L}{\partial \dot x^i \partial \dot x^j}\frac {\partial x^i}{\partial q^k}\frac {\partial x^j}{\partial q^l}\right )_{k,l=1,\ldots ,n} \neq 0. \end{equation}

As before we can define the constrained motion by minimizing the corresponding action restricted on the space of curves in \(Q\) with the appropriate fixed endpoints in \(Q\). Due to the immersion of \(Q\) into \(M\), which in coordinates reads \(x^1 = x^1(q)\), \(\ldots \), \(x^m=x^m(q)\), and the corresponding immersion of tangent bundles, we can define the restriction of the lagrangian \(L\) on the subbundle \(TQ\):

\begin{equation} \label {eq:dalembertmanifold} L_Q(q,\dot q) = L\left (x(q), \frac {\partial x}{\partial q}\dot q\right ). \end{equation}