Editing Analytical mechanics (section)

==Properties of the Lagrangian and the Hamiltonian==
Following are overlapping properties between the Lagrangian and Hamiltonian functions.<ref name="autogenerated1"/><ref>''Classical Mechanics'', T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{ISBN|0-07-084018-0}}</ref>
* All the individual generalized coordinates ''q<sub>i</sub>''(''t''), velocities ''q̇<sub>i</sub>''(''t'') and momenta ''p<sub>i</sub>''(''t'') for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time ''t'' as a variable in addition to the '''q'''(''t''), '''p'''(''t''), not simply as a parameter through '''q'''(''t'') and '''p'''(''t''), which would mean explicit time-independence.
* The Lagrangian is invariant under addition of the ''[[total derivative|total]]'' [[time derivative]] of any function of '''q''' and ''t'', that is: <math display="block">L' = L +\frac{d}{dt}F(\mathbf{q},t) \,,</math> so each Lagrangian ''L'' and ''L''' describe ''exactly the same motion''. In other words, the Lagrangian of a system is not unique.
* Analogously, the Hamiltonian is invariant under addition of the ''[[partial derivative|partial]]'' time derivative of any function of '''q''', '''p''' and ''t'', that is: <math display="block">K = H + \frac{\partial}{\partial t}G(\mathbf{q},\mathbf{p},t) \,,</math> (''K'' is a frequently used letter in this case). This property is used in [[canonical transformations]] (see below).
*If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are [[Constant of motion|constants of the motion]], i.e. are [[conserved quantity|conserved]], this immediately follows from Lagrange's equations: <math display="block">\frac{\partial L}{\partial q_j }=0\,\rightarrow \,\frac{dp_j}{dt} = \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j}=0 </math> Such coordinates are "[[Lagrangian mechanics|cyclic]]" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.
*If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).
*If the kinetic energy is a [[homogeneous function]] of degree 2 of the generalized velocities, ''and'' the Lagrangian is explicitly time-independent, then: <math display="block">T((\lambda \dot{q}_i)^2, (\lambda \dot{q}_j \lambda \dot{q}_k), \mathbf{q}) = \lambda^2 T((\dot{q}_i)^2, \dot{q}_j\dot{q}_k, \mathbf{q})\,,\quad L(\mathbf{q},\mathbf{\dot{q}})\,,</math> where ''λ'' is a constant, then the Hamiltonian will be the ''total conserved energy'', equal to the total kinetic and potential energies of the system: <math display="block">H = T + V = E\,.</math> This is the basis for the [[Schrödinger equation]], inserting [[operators (physics)|quantum operators]] directly obtains it.