Abstract
Canonical mechanics is a central part of general mechanics, where one goes beyond the somewhat narrow framework of Newtonian mechanics with position coordinates in the three-dimensional space, towards a more general formulation of mechanical systems belonging to a much larger class.
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Notes
- 1.
Here we make use of this somewhat archaic but very intuitive notion. Geometrically speaking, virtual displacements are described by tangent vectors of the smooth hypersurface in \(\mathbb {R}^{3n}\) that is defined by (2.4). D’Alembert’s principle can and should be formulated in the geometric framework of Chap. 5.
- 2.
The name action arises because L has the dimension of energy: the product (energy \(\times \) time) is called action, and this is indeed the dimension of the action integral.
- 3.
- 4.
In mechanics the kinetic energy and, hence, the Lagrangian are positive-definite (but not necessarily homogeneous) quadratic functions of the variables In this situation, solving the defining equations for \(p_k\) in terms of the \(q_i\) yields a unique solution also globally.
- 5.
See the precise definition in Sect. 5.5.4 below.
- 6.
The derivative of H by \(x_k\) is written as \(H_{, x_k}\). More generally, the set of all derivatives of H by is abbreviated by \(H_{, x}\).
- 7.
We define the bracket such that it corresponds to the commutator [f, g] of quantum mechanics, without change of sign.
- 8.
This is a special case of the more general rectification theorem for general, autonomous, differentiable systems: in the neighborhood of any point that is not an equilibrium position (i.e. where ), the system of first-order differential equations can be transformed to the form , i.e. \(\dot{z}_1=1\), \(\dot{z}_2=0=\ldots =\dot{z}_f\). For a proof see e.g. Arnol’d (1973).
- 9.
There is evidence, from numerical studies, that the motion of the planet Pluto is chaotic, i.e. that it is intrinsically unstable over large time scales (G.J. Sussman and J. Wisdom , Science 241 (1988) 433). Because Pluto couples to the other planets, though weakly, this irregular behavior eventually spreads to the whole system.
- 10.
Equation (2.175) holds for functions \(f_{\varvec{k}}=\exp \{\mathrm{i}\sum k_i \theta _i(t)\}\) with \(\theta _i(t)=\omega _it+\beta _i\), where it gives, in fact, \(\langle f_{\varvec{k}} \rangle =0\), except for \(k_1=\cdots =k_f=0\). Any continuous F can be approximated by a finite linear combination \(F=\sum C_{\varvec{k}} f_{\varvec{k}}\).
- 11.
If, in turn, the frequencies are rationally dependent, the tori are said to be resonant tori , cf. Example (vi) of Sect. 2.37.2. In this case the motion is quasiperiodic with a number of frequencies that is smaller than f.
- 12.
W. Sarlet, F. Cantrijn; SIAM Review 23 (1981) 467.
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Appendix: Practical Examples
Appendix: Practical Examples
1. Small Oscillations. Let a Lagrangian system be described in terms of f generalized coordinates \(\{ q_i\}\), each of which can oscillate around an equilibrium position \(q_i^0\). The potential energy \(U(q_1,\ldots , q_f)\) having an absolute minimum \(U_0\) at \((q_1^0,\ldots , q_f^0)\) one may visualize this system as a lattice defined by the equilibrium positions \((q_1^0,\ldots , q_f^0)\), the edges of which can oscillate around this configuration. The limit of small oscillations is realized if the potential energy can be approximated by a quadratic form in the neighborhood of its minimum, viz.
Note that for the mathematical pendulum (which has \(f=1\)) this is identical with the limit of small deviations from the vertical, i.e. the limit of harmonic oscillation. For \(f>1\) this is a system of coupled harmonic oscillators.
Derive the equations of motion and find the normal modes of this system.
Solution. It is clear that only the symmetric part of the coefficients \(u_{ik}\) is dynamically relevant, \(a_{ik}{\;\mathop {=}\limits ^\mathrm{def}\;}(u_{ik}+u_{ki})/2\). As U has a minimum, the matrix
is not only real and symmetric but also positive. This means that all its eigenvalues are real and positive-semidefinite. It is useful to replace the variables \(q_i\) by the deviations from equilibrium, \(z_i = q_i-q_i^0\). The kinetic energy is a quadratic form of the time derivatives of \(q_i\) or, equivalently of \(z_i\), with symmetric coefficients:
The matrix \(\{t_{ik}\}\) is not singular and is positive as well. Therefore, one can choose the natural form for the Lagrangian function
from which follows the system of coupled equations
For \(f=1\) this is the equation of the harmonic oscillator. This suggests solving the general case by means of the substitution
The complex form is chosen in order to simplify the calculations. In the end we shall have to take the real part of the eigenmodes. Inserting this expression for z into the equations of motion (A.3) yields the following system of coupled linear equations:
This has a nontrivial solutions if and only if the determinant of its coefficient vanishes,
This equation has f positive-semidefinite solutions
which are said to be the eigenfrequencies of the system.
As an example we consider two identical harmonic oscillators (frequency \(\omega _0\)) that are coupled by means of a harmonic spring. The spring is not active when both oscillators are at rest (or, more generally, whenever the difference of their positions is the same as at rest). It is not difficult to guess the eigenfrequencies of this system: (i) the two oscillators swing in phase, the spring remains inactive; (ii) the oscillators swing in opposite phase. Let us verify this behavior within the general analysis. We have
Taking out the common factor m, the system (A.4) reads
The condition (A.5) yields a quadratic equation whose solutions are
Inserting these, one by one, into the system of equations (A.4′), one finds
(The normalization is free. We choose \(a^{(i)}_1=1/\sqrt{2}\), \(i=1,2\)). Thus, we indeed obtain the expected solutions. The linear combinations above, i.e.
decouple the system completely. The Lagrangian function becomes
It describes two independent linear oscillators. The new variables \(Q_i\) are said to be normal coordinates of the system. They are defined by the eigenvectors of the matrix \((a_{ij}-\varOmega ^2_lt_{ij})\) and correspond to the eigenvalues \(\varOmega ^2_l\).
In the general case \((f>2)\) one proceeds in an analogous fashion. Determine the frequencies from (A.5) and insert them, one by one, into (A.4). Solve this system and determine the eigenvectors \((a_1^{(l)},\ldots , a_f^{(l)})\) (up to normalization) that pertain to the eigenvalues \(\varOmega ^2_l\).
If all eigenvalues are different, the eigenvectors are uniquely determined up to normalization. We write (A.4) for two different eigenvalues,
and multiply the first equation by \(a_i^{(p)}\) from the left, the second by \(a_j^{(q)}\) from the left. We sum the first over i and the second over j and take their difference. Both \(t_{ij}\) and \(a_{ij}\) are symmetric. Therefore, we obtain
As \(\varOmega ^2_p\ne \varOmega ^2_q\), the double sum must vanish if \(p\ne q\). For \(p=q\), we can normalize the eigenvectors such that the double sum gives 1. We conclude that
Equation (A.6a) and the result above can be combined to obtain
This result tells us that the matrices \(t_{ij}\) and \(a_{ij}\) are diagonalized simultaneously. We then set
and insert this into the Lagrangian function to obtain
Thus, we have achieved the transformation to normal coordinates.
If some of the frequencies are degenerate, the corresponding eigenvectors are no longer uniquely determined. It is always possible, however, to choose s linearly independent vectors in the subspace that belongs to \(\varOmega _{r_1}=\varOmega _{r_2}=\cdots =\varOmega _{r_s}\) (s denotes the degree of degeneracy). This construction is given in courses on linear algebra.
One can go further and try several examples on a PC: a linear chain of n oscillators with harmonic couplings, a planar lattice of mass points joined by harmonic springs, etc., for which the matrices \(t_{ik}\) and \(a_{ik}\) are easily constructed. If one has at one’s disposal routines for matrix calculations, it is not difficult to find the eigenfrequencies and the normal coordinates.
2. The Planar Mathematical Pendulum and Liouville’s Theorem. Work out (numerically) Example (ii) of Sect. 2.30 and illustrate it with some figures.
Solution. We follow the notation of Sect. 1.17.2, i.e. we take \(z_1=\varphi \) as the generalized coordinate and \(z_2=\dot{\varphi }/\omega \) as the generalized momentum, where \(\omega =\sqrt{g/l}\) is the frequency of the corresponding harmonic oscillator and \(\tau =\omega t\). Thus, time is measured in units of \((\omega )^{-1}\), The energy is measured in units of mgl, i.e.
\(\varepsilon \) is positive-semidefinite. \(\varepsilon <2\) pertains to the oscillating solutions, \(\varepsilon =2\) is the separatrix, and \(\varepsilon >2\) pertains to the rotating solutions. The equations of motion (1.40) yield the second-order differential equation for \(z_{1}\)
First, one verifies that \(z_1\) and \(z_2\) are indeed conjugate variables, provided one uses \(\tau \) as time variable. In order to see this start from the dimensionless Lagrangian function
and take its derivative with respect to \(\dot{z}_1=(\mathrm{d}z_1/\mathrm{d}\tau )\). This gives \(z_2=(\mathrm{d}z_1/\mathrm{d}\tau )=(\mathrm{d}\varphi /\mathrm{d} t)/\omega \), as expected.
For drawing the phase portraits, Fig. 1.10, it is sufficient to plot \(z_2\) as a function of \(z_1\), as obtained from (A.9). This is not sufficient, however, if we wish to follow the motion along the phase curves, as a function of time. As we wish to study the time evolution of an ensemble of initial conditions, we must integrate the differential equation (A.10). This integration can be done numerically, e.g. by means of a Runge–Kutta procedure (cf. Abramowitz and Stegun 1965, Sect. 25.5.22). Equation (A.9) has the form \(y^{\prime \prime }=-\sin y\). Let h be the step size and \(y_n\) and \(y_n^\prime \) the values of the function and its derivative respectively at \(\tau _n\). Their values at \(\tau _{n+1}=\tau _n+h\) are obtained by the following series of steps. Let
Then
Note that y is our \(z_1\) while \(y^\prime \) is \(z_2\) and that the two are related by (A.9) to the reduced energy \(\varepsilon \). Equations (A.12) are easy to implement on a computer. Choose an initial configuration \((y_0=z_1(0)\), \(y^\prime _0=z_2(0)\)), take \(h=\pi /30\), for example, and run the program until the time variable has reached a given endpoint \(\tau \). Using the dimensionless variable \(\tau \), the harmonic oscillator (corresponding to small oscillations of the pendulum) has the period \(T^{(0)} =2\pi \). It is convenient, therefore, to choose the end point to be \(T^{(0)}\) or fractions thereof. This shows very clearly the retardation of the pendulum motion as compared to the oscillator: points on pendulum phase portraits with \(0<\varepsilon \ll 2\) move almost as fast as points on the oscillator portrait; the closer \(\varepsilon \) approaches 2 from below, the more they are retarded compared to the oscillator. Points on the separatrix \((\varepsilon =2)\) that start from, say, \((z_1=0,z_2=2)\) can never move beyond the first quadrant of the \((z_1,z_2)\)-plane. They approach the point \((\pi , 0)\) asymptotically, as \(\tau \) goes to infinity.
In the examples shown in Figs. 2.13, 2.14 and 2.15 we study the flow of an initial ensemble of 32 points on a circle with radius \(r=0.5\) and the center of that circle, for the time intervals indicated in the figures. This allows one to follow the motion of each individual point. As an example, in Fig. 2.14 we have marked with arrows the consecutive positions of the point that started from the configuration (0, 1).
Of course, one may try other shapes for the initial ensemble (instead of the circle) and follow its flow through phase space. A good test of the program is to replace the right-hand side of (A.10) with \(-z_1\). This should give the picture shown in Fig. 2.12.
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Scheck, F. (2018). The Principles of Canonical Mechanics. In: Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55490-6_2
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