Classical Mechanics¶

Pysic-rs implements the two variational formulations of mechanics — Hamiltonian and Lagrangian — plus rigid-body dynamics with the full suite of rotation parameterisations. Everything here is textbook (proven) mechanics.

Python binding status: euler_equations, inertia_tensor, and euler_to_rotation are bound. Hamiltonian/Lagrangian picture, Poisson brackets, and torque_free are Rust-only. See the API page.

1. Lagrangian Mechanics¶

1.1 Configuration Space & Velocity¶

A system with \(n\) generalised coordinates \(q = (q_1,\ldots,q_n)\) has configuration space \(Q\) (a manifold). The velocity is \(\dot{q} = (\dot{q}_1,\ldots,\dot{q}_n)\).

1.2 The Lagrangian¶

\[ L(q,\dot{q},t) = T - V \]

where \(T = \frac{1}{2}\sum_{ij}m_{ij}(q)\dot{q}_i\dot{q}_j\) is the kinetic energy (may be non-diagonal in curvilinear coordinates) and \(V(q,t)\) is the potential.

1.3 Hamilton’s Principle¶

Theorem (principle of least action). The physical trajectory between fixed endpoints \(q(t_1)=q_1\), \(q(t_2)=q_2\) extremises the action:

\[ S[q] = \int_{t_1}^{t_2}L(q,\dot{q},t)\,dt \]

Proof. Consider a variation \(q\to q+\delta q\) with \(\delta q(t_1)=\delta q(t_2)=0\):

\[ \delta S = \int_{t_1}^{t_2}\left(\frac{\partial L}{\partial q_i}\delta q_i + \frac{\partial L}{\partial\dot{q}_i}\delta\dot{q}_i\right)dt \]

Integrate the second term by parts:

\[ \delta S = \int_{t_1}^{t_2}\left[\frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_i}\right]\delta q_i\,dt + \left[\frac{\partial L}{\partial\dot{q}_i}\delta q_i\right]_{t_1}^{t_2} \]

The boundary term vanishes. For \(\delta S=0\) for all \(\delta q_i\), each bracket must vanish: the Euler–Lagrange equation. \(\square\)

1.4 Euler–Lagrange Equation¶

\[ \boxed{\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_i} - \frac{\partial L}{\partial q_i} = 0, \qquad i=1,\ldots,n} \]

1.5 Noether’s Theorem¶

Theorem (Noether, 1918). Every continuous symmetry of the Lagrangian corresponds to a conserved quantity.

Proof. Under a one-parameter transformation \(q_i\to q_i+\varepsilon\,Q_i(q,\dot{q})\) with \(\delta L = 0\):

\[ 0 = \frac{\partial L}{\partial q_i}Q_i + \frac{\partial L}{\partial\dot{q}_i}\dot{Q}_i = \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_i}Q_i\right) \]

Hence \(J = \frac{\partial L}{\partial\dot{q}_i}Q_i\) is conserved. \(\square\)

Examples:

  • Time translation → energy conservation.

  • Spatial translation → momentum conservation.

  • Rotation → angular momentum conservation.

2. Hamiltonian Mechanics¶

2.1 Legendre Transform¶

The canonical momenta are

\[ p_i = \frac{\partial L}{\partial\dot{q}_i} \]

The Hamiltonian is the Legendre transform of \(L\):

\[ H(q,p,t) = \sum_i p_i\dot{q}_i - L(q,\dot{q},t) \]

For natural systems (\(T\) quadratic in \(\dot{q}\), \(V\) independent of \(\dot{q}\)): \(H = T + V\) = total energy.

2.2 Hamilton’s Canonical Equations¶

\[ \boxed{\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}} \]

Proof. Differentiate \(H = \sum p_i\dot{q}_i - L\):

\[ \frac{\partial H}{\partial p_i} = \dot{q}_i + \sum_j p_j\frac{\partial\dot{q}_j}{\partial p_i} - \sum_j\frac{\partial L}{\partial\dot{q}_j}\frac{\partial\dot{q}_j}{\partial p_i} = \dot{q}_i + \sum_j\left(p_j-\frac{\partial L}{\partial\dot{q}_j}\right)\frac{\partial\dot{q}_j}{\partial p_i} = \dot{q}_i \]

since \(p_j = \partial L/\partial\dot{q}_j\). Similarly for \(\partial H/\partial q_i\). \(\square\)

2.3 Poisson Bracket¶

\[ \{f,g\} = \sum_i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right) \]

Properties (proven — Lie algebra structure):

  • Bilinearity, antisymmetry: \(\{f,g\}=-\{g,f\}\).

  • Jacobi identity: \(\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0\).

  • Leibniz rule: \(\{fg,h\}=f\{g,h\}+g\{f,h\}\).

  • Time evolution: \(\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}\).

An observable with \(\{f,H\}=0\) (and \(\partial f/\partial t=0\)) is a constant of motion.

2.4 Canonical Transformations¶

A transformation \((q,p)\to(Q,P)\) is canonical if it preserves the Poisson bracket structure:

\[ \{Q_i,P_j\}_{\rm new} = \delta_{ij}, \qquad \{Q_i,Q_j\}_{\rm new}=0, \qquad \{P_i,P_j\}_{\rm new}=0 \]

Equivalently (proven): the transformation preserves the symplectic form \(\Omega = \sum_i dq_i\wedge dp_i\).

3. Rigid Body Dynamics¶

3.1 Inertia Tensor¶

For point masses \(m_a\) at positions \(\mathbf{r}_a\):

\[ \boxed{I_{ij} = \sum_a m_a\left(\|\mathbf{r}_a\|^2\delta_{ij} - r_{a,i}r_{a,j}\right)} \]

Properties:

  • Symmetric: \(I_{ij}=I_{ji}\).

  • Positive semi-definite.

  • Diagonalisable in the principal-axis frame: \(I = \mathrm{diag}(I_1,I_2,I_3)\).

3.2 Euler’s Equations¶

In the body frame (rotating with angular velocity \(\boldsymbol{\omega}\)), with principal moments \(I_1,I_2,I_3\):

\[ \boxed{I_1\dot{\omega}_1 = (I_2-I_3)\omega_2\omega_3 + N_1} \]
\[ I_2\dot{\omega}_2 = (I_3-I_1)\omega_3\omega_1 + N_2 \]
\[ I_3\dot{\omega}_3 = (I_1-I_2)\omega_1\omega_2 + N_3 \]

Proof. From \(\dot{\mathbf{L}} = \mathbf{N}\) in the lab frame, and \(\dot{\mathbf{L}}_{\rm body} = \mathbf{N} - \boldsymbol{\omega}\times\mathbf{L}\) (the transport theorem). In the principal frame, \(\mathbf{L} = (I_1\omega_1,I_2\omega_2, I_3\omega_3)\), and the cross product gives the nonlinear coupling terms. \(\square\)

3.3 Torque-Free Motion¶

For \(\mathbf{N}=0\), both kinetic energy \(T = \frac{1}{2}\sum I_i\omega_i^2\) and angular momentum magnitude \(|\mathbf{L}|\) are conserved. The tip of \(\boldsymbol{\omega}\) traces an ellipse on the inertia ellipsoid.

Symmetric top (\(I_1=I_2\)): closed-form solution

\[ \omega(t) = \bigl(A\cos(\Omega t),\,A\sin(\Omega t),\,\omega_3\bigr), \qquad \Omega = \frac{I_3-I_1}{I_1}\omega_3 \]

This is free precession — the angular velocity vector rotates around the symmetry axis.

4. Routines¶

Routine

Description

hamiltonian_h(T, V)

\(H=T+V\)

canonical_equations(H, q, p, h)

\(\dot{q},\dot{p}\) pairs

poisson_bracket(f, g, q, p, h)

\(\{f,g\}\)

canonical_transform(...)

symplectic-coordinate check

lagrangian_l(T, V)

\(L=T-V\)

euler_lagrange(L, q, qdot, h)

EL residual

kinetic_energy(m, v), potential_energy(...)

\(T\), \(V\)

euler_equations(I, omega, torque)

\(\dot{\boldsymbol{\omega}}\)

torque_free(I, omega0, dt, steps)

free precession

inertia_tensor(masses, positions)

\(I_{ij}\)

angular_momentum(I, omega)

\(\mathbf{L}\)

5. Usage Examples¶

Harmonic oscillator via canonical equations¶

from pysicrs import canonical_equations, hamiltonian_h

H = lambda q, p: 0.5*p**2 + 0.5*q**2
print(canonical_equations(H, 0.0, 1.0, 1e-5))  # (1.0, ~0.0)

Free symmetric top¶

from pysicrs import torque_free

I = [2.0, 2.0, 1.0]
omega0 = [1.0, 0.0, 2.0]
traj = torque_free(I, omega0, dt=0.01, steps=1000)
print(len(traj), traj[-1])  # bounded periodic nutation

Angular momentum of a rotating body¶

from pysicrs import inertia_tensor, angular_momentum

I = inertia_tensor(masses=[1.0, 1.0],
                   positions=[[1.0, 0.0, 0.0], [-1.0, 0.0, 0.0]])
L = angular_momentum(I, [0.0, 0.0, 2.0])
print(L)  # about áş‘

6. Advantages & Limitations¶

✅ Both formulations exported — match your problem’s natural coordinates

âś… Symplectic integrators (ODE) preserve the bracket structure for long times

âś… Rigid-body covers free tops, precession and torque-driven rotation

❌ No constraint mechanics (holonomic constraints not solved directly)

❌ Finite-difference derivatives need a well-chosen step

❌ No field-theoretic (continuum) classical mechanics yet

7. References¶

  1. Goldstein, H., Poole, C. & Safko, J. (2002). Classical Mechanics, 3rd ed. Addison-Wesley.

  2. Landau, L.D. & Lifshitz, E.M. (1976). Mechanics, 3rd ed. Butterworth-Heinemann.

  3. Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, 2nd ed. Springer.

  4. Hairer, E., Lubich, C. & Wanner, G. (2006). Geometric Numerical Integration, 2nd ed. Springer.