Deep Research

Hamiltonian Structures: Dimensional Analysis of Conjugate Variables

Survey of Hamiltonian structures across physics, analyzing the dimensions of conjugate variable pairs in different physical systems.

The fundamental requirement for a Hamiltonian system is that the conjugate variables (q,p)(q, p) satisfy:

[q]⋅[p]=action=kg⋅m2⋅s−1[q] \cdot [p] = \textrm{action} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1}

This ensures the Hamiltonian HH always has dimensions of energy, and Hamilton's equations produce physically meaningful time derivatives.

Case 1: Angle and Angular Momentum

Conjugate pair:

  • Position: θ\theta (angle), dimensionless
  • Momentum: LL (angular momentum)

Dimensional check that [q]⋅[p][q] \cdot [p] = action:

[L]=kg⋅m2⋅s−1[L] = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1}

[θ]⋅[L]=1⋅kg⋅m2⋅s−1=kg⋅m2⋅s−1✓[\theta] \cdot [L] = 1 \cdot \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \quad ✓

Hamiltonian (free symmetric rotor with moment of inertia II):

H=L22IH = \frac{L^2}{2I}

where [I]=kg⋅m2[I] = \textrm{kg} \cdot \textrm{m}^2, giving [H]=kg⋅m2⋅s−2[H] = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2} (energy) ✓

Hamilton's equations:

θ˙=∂H∂L=LI=ω\dot{\theta} = \frac{\partial H}{\partial L} = \frac{L}{I} = \omega

This is the angular velocity.

L˙=−∂H∂θ=0\dot{L} = -\frac{\partial H}{\partial \theta} = 0

Or −τ-\tau if a potential is present.

Dimensional verification:

Differentiating HH with respect to LL:

[H][L]=kg⋅m2⋅s−2kg⋅m2⋅s−1=s−1=[ω]✓\frac{[H]}{[L]} = \frac{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2}}{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1}} = \textrm{s}^{-1} = [\omega] \quad ✓

Differentiating HH with respect to θ\theta:

[H][θ]=kg⋅m2⋅s−2=[τ]✓\frac{[H]}{[\theta]} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2} = [\tau] \quad ✓

The torque dimensions are correct since [τ]=[r×F]=m⋅kg⋅m⋅s−2=kg⋅m2⋅s−2[\tau] = [r \times F] = \textrm{m} \cdot \textrm{kg} \cdot \textrm{m} \cdot \textrm{s}^{-2} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2}.

Case 2: Charge and Magnetic Flux

Conjugate pair:

  • Position: QQ (charge), with [Q]=C[Q] = \textrm{C}
  • Momentum: Φ\Phi (magnetic flux)

Flux in base units: Starting from Wb = V·s and V = J/C:

[Φ]=kg⋅m2⋅s−2⋅C−1⋅s=kg⋅m2⋅s−1⋅C−1[\Phi] = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2} \cdot \textrm{C}^{-1} \cdot \textrm{s} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \cdot \textrm{C}^{-1}

Dimensional check:

[Q]⋅[Φ]=C⋅kg⋅m2⋅s−1⋅C−1=kg⋅m2⋅s−1✓[Q] \cdot [\Phi] = \textrm{C} \cdot \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \cdot \textrm{C}^{-1} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \quad ✓

Hamiltonian (LC circuit):

H=Φ22L+Q22CH = \frac{\Phi^2}{2L} + \frac{Q^2}{2C}

where LL is inductance and CC is capacitance. Their base units are:

[L]=kg⋅m2⋅C−2[L] = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{C}^{-2}

[C]=C2⋅s2⋅kg−1⋅m−2[C] = \textrm{C}^2 \cdot \textrm{s}^2 \cdot \textrm{kg}^{-1} \cdot \textrm{m}^{-2}

You can verify each term has energy dimensions:

[Φ]2[L]=(kg⋅m2⋅s−1⋅C−1)2kg⋅m2⋅C−2=kg⋅m2⋅s−2✓\frac{[\Phi]^2}{[L]} = \frac{(\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \cdot \textrm{C}^{-1})^2}{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{C}^{-2}} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2} \quad ✓

[Q]2[C]=C2C2⋅s2⋅kg−1⋅m−2=kg⋅m2⋅s−2✓\frac{[Q]^2}{[C]} = \frac{\textrm{C}^2}{\textrm{C}^2 \cdot \textrm{s}^2 \cdot \textrm{kg}^{-1} \cdot \textrm{m}^{-2}} = \textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2} \quad ✓

Hamilton's equations:

Q˙=∂H∂Φ=ΦL=I\dot{Q} = \frac{\partial H}{\partial \Phi} = \frac{\Phi}{L} = I

This is the current.

Φ˙=−∂H∂Q=−QC=−V\dot{\Phi} = -\frac{\partial H}{\partial Q} = -\frac{Q}{C} = -V

This is the negative voltage.

Dimensional verification:

Differentiating HH with respect to Φ\Phi:

[H][Φ]=kg⋅m2⋅s−2kg⋅m2⋅s−1⋅C−1=C⋅s−1=A✓\frac{[H]}{[\Phi]} = \frac{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2}}{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-1} \cdot \textrm{C}^{-1}} = \textrm{C} \cdot \textrm{s}^{-1} = \textrm{A} \quad ✓

Differentiating HH with respect to QQ:

[H][Q]=kg⋅m2⋅s−2C=V✓\frac{[H]}{[Q]} = \frac{\textrm{kg} \cdot \textrm{m}^2 \cdot \textrm{s}^{-2}}{\textrm{C}} = \textrm{V} \quad ✓

Physical interpretation: The first equation Q˙=I\dot{Q} = I is just the definition of current. The second equation Φ˙=−V\dot{\Phi} = -V is Faraday's law applied around the LC loop: the capacitor voltage V=Q/CV = Q/C drives flux change in the inductor. The minus sign reflects Kirchhoff's voltage law (VC+VL=0V_C + V_L = 0 around the loop).

Summary

| System | qq | pp | [q][q] | [p][p] | q˙\dot{q} | −p˙-\dot{p} | |-----------------|----------|--------|--------|-----------------|------------|------------| | Mechanical | xx | pp | m | kg·m·s⁻¹ | velocity | force | | Rotational | θ\theta | LL | 1 | kg·m²·s⁻¹ | ang. vel. | torque | | Electromagnetic | QQ | Φ\Phi | C | kg·m²·s⁻¹·C⁻¹ | current | voltage |

The deep unity here is that all three systems share the same symplectic structure—they are different physical realizations of the same abstract phase space geometry. The canonical 2-form ω=dq∧dp\omega = dq \wedge dp is preserved under time evolution in each case, which is ultimately why energy is conserved and why these systems exhibit their characteristic oscillatory behavior.