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%% 4.1   Classical mechanics
% Trajectories
% Classical mechanics deals with objects. The objects are characterized by a name or pointer, and by context-independent properties. Object properties are first the mass m and its position $\vec r$. The position changes continuously over time, resulting in the trajectory \vec r\left( t \right) and velocity $\vec v = \dot \vec r$.
% Content and dynamics are united by the requirement that for each point of the trajectory the velocity of the object is proportional to the momentum. Thus, the classical momentum becomes a function of time.
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% (4‑1)
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% A trajectory is laid over the momentum field so that the velocity
% vector is proportional to the momentum vector everywhere
% In (4.1), the trajectory results from an integral equation which is generally difficult to solve. Usually, the trajectory is determined by a differential equation.
% Newtonian equation of motion from energy conservation
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% (4‑2)
% These equations can then be solved analytically or numerically.
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% Separation of the variables and integration of the momentum
% In integrable systems, the separation of the individual degrees of freedom is achieved by selecting suitable coordinates. The trajectory can then be determined by solving one-dimensional integrals.
%
% (4‑3)
%
% (4‑4)
% Hamilton’s  equations (HE)
% This is the most general representation of classical physics. It also allows us to determine 'trajectories' for generalized coordinates and non-integrable systems. The Hamiltonian H is introduced as a field of generalized coordinates and momenta. If the function value of H  is constantly equal to the energy, the space coordinates and momenta are implicitly linked. The additional demand for \[{\dot q^i} = \frac{1}{m}{g^{ij}}{p_j}\] on the trajectory leads to the HE.
%
% (4‑5)
% On a trajectory, the Hamiltonian is equal to the energy, that is, constant.
% Thereby it also follows the momentum change on the trajectory:
%
% (4‑6)
% This is the popular formulation of classical mechanics:
%
% (4‑7)
% Cyclic variables
% If the Hamiltonian does not depend on a coordinate ${q^i}$, then the conjugate momentum ${p_i}$ is a conserved quantity, that is a constant of the motion.
%
% (4‑8)
% Generalization
% All possible forms of classical systems can be handled by choosing suitable coordinates .
%
% (4‑9)