06_27_2025: updated docs

Author: i-a-morozov

docs/source/index.rst

@@ -6,20 +6,26 @@ Single particle hamiltonian function:
 
 .. math::
 
-   H(q_x, q_y, q_s, p_x, p_y, p_s; s) = p_s/\beta - t(s)(q_x p_y - q_y p_x) - (1 + h(s) q_x) (\sqrt{(p_s + 1/beta - \varphi(qs, s))^2 - (p_x - a_x(qs, s))^2 - (p_y - a_y(qs, s))^2 - 1/(\beta \gamma)^2} - a_s)
+   H = H(q_x, q_y, q_s, p_x, p_y, p_s; s) \\
+   H = p_s/\beta - t(s)(q_x p_y - q_y p_x) - (1 + h(s) q_x) (\sqrt{P_s^2 - P_x^2 - P_y^2 - 1/(\beta \gamma)^2} - a_s)  \\
+   P_s = p_s + 1/beta - \varphi(q_x, q_y, q_s, s)  \\
+   P_x = p_x - a_x(q_x, q_y, q_s, s)  \\
+   P_y  p_y - a_y(q_x, q_y, q_s, s)
 
 where :math:`\beta` and :math:`\gamma` are the relativistic factors, :math:`h(s)` is the reference trajectory curvature and :math:`t(s)` is the reference trajectory torsion, :math:`a_x(qs, s)`, :math:`a_y(qs, s)` and :math:`a_s(qs, s)` are the scaled vector potential components, and :math:`\varphi(qs, s)` is the scaled scalar potential.
 Additionaly, longitudinal coordinate and momentum are given by:
 
 .. math::
-   q_s = \frac{s}{\beta} - c t
+
+   q_s = \frac{s}{\beta} - c t \\
    p_s = \frac{E}{c P} - 1/beta
 
 Common predefined elements are available or you can create your own by specifying scaled potentials and reference trajectory parameters (curvature and torsion).
 All but vector potentials arguments are optional for hamiltonian and element construction.
 Vector and scalar potentials are assumed to have matching signatures.
 
 .. code-block:: python
+
    def vector(qs:Array, s:Array, *args:Array) -> tuple[Array, Array, Array]:
       q_x, q_y, q_s = qs
       ...
@@ -32,6 +38,7 @@ This is also the case for curvature and torsion functions.
 Note, same extra arguments as in vector and scalar functions should be passed.
 
 .. code-block:: python
+
    def curvature(s:Array, *args:Array) -> Array:
       ...
 
@@ -41,6 +48,7 @@ Note, same extra arguments as in vector and scalar functions should be passed.
 The resulting hamiltonian and element signatures are:
 
 .. code-block:: python
+
    def hamiltonian(qs: Array, ps: Array, s: Array, *args: Array) -> Array:
       q_x, q_y, q_s = qs
       p_x, p_y, p_s = ps