Hamiltonian formulation of classical fields with fractional derivatives: revisited

Abstract

An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton’s equations are obtained for two classical field examples. The formulation presented and the resulting equations are very similar to those appearing in classical field theory.

Appendix: Variation of full Hamiltonian

We can rewrite Eq. (27) as:

$$ \begin{aligned}[b] H =& \int \bigl[ \pi_{\alpha} {}_{a}D_{t}^{\alpha} \psi+ \pi_{\alpha}^{ *} {}_{a}D_{t}^{\alpha} \psi^{ *} + \pi_{\beta} {}_{t}D_{b}^{\beta} \psi \\ &{}+ \pi_{\beta}^{ *} {}_{t}D_{b}^{\beta} \psi^{ *} \bigr]\, d^{3}r - L. \end{aligned} $$

(A.1)

Now, take the variation of H, we get:

$$ \begin{aligned}[b] \delta H =& \int\delta\bigl(\pi_{\alpha} {}_{a}D_{t}^{\alpha} \psi+ \pi_{\beta} {}_{t}D_{b}^{\beta} \psi \bigr) \,d^{3}r \\ &{}+ \int\delta\bigl(\pi_{\alpha}^{ *} {}_{a}D_{t}^{\alpha} \psi^{ *} + \pi_{\beta}^{ *} {}_{t}D_{b}^{\beta} \psi^{ *} \bigr) \,d^{3}r - \delta L. \\[6pt] \end{aligned} $$

(A.2)

Using Eq. (19), Eq. (21) and Eq. (23), we rewrite the variation of full Lagrangian given by Eq. (17) as:

$$ \begin{aligned}[b] \delta L =& \int\bigl\{ - \bigl({}_{a}D_{t}^{\beta} \pi_{\beta} + {}_{t}D_{b}^{\alpha} \pi_{\alpha} \bigr) \delta\psi+ \pi_{\alpha} \delta \bigl({}_{a}D_{t}^{\alpha} \psi\bigr) \\ &{}+ \pi_{\beta} \delta\bigl({}_{t}D_{b}^{\beta} \psi\bigr)\bigr\} \,d^{3}r. \end{aligned} $$

(A.3)

The above equation can be arranged as:

$$ \begin{aligned}[b] \delta L =& \int\bigl\{ - \bigl({}_{a}D_{t}^{\beta} \pi_{\beta} + {}_{t}D_{b}^{\alpha} \pi_{\alpha} \bigr) \delta\psi \\ &{}+ \delta\bigl(\pi_{\alpha} {}_{a}D_{t}^{\alpha} \psi+ \pi_{\beta} {}_{t}D_{b}^{\beta} \psi\bigr). \\ &{}- {}_{a}D_{t}^{\alpha} \psi\delta\pi_{\alpha} - {}_{t}D_{b}^{\beta} \psi \delta\pi_{\beta} \bigr\} \,d^{3}r. \end{aligned} $$

(A.4)

Substituting Eq. (A.4) into Eq. (A.2), one gets:

(A.5)

Taking the time integration of δH, we get:

(A.6)

Integrate (by parts) the last two terms in the above equation then separate integrals over time and integrals over space to get:

(A.7)