Editing Analytical mechanics (section)

==Hamiltonian mechanics==
{{main | Hamiltonian mechanics }}

The [[Legendre transformation]] of the Lagrangian replaces the generalized coordinates and velocities ('''q''', '''q̇''') with ('''q''', '''p'''); the generalized coordinates and the ''[[Canonical coordinates|generalized momenta]]'' conjugate to the generalized coordinates:

:<math>\mathbf{p} = \frac{\partial L}{\partial \mathbf{\dot{q}}} = \left(\frac{\partial L}{\partial \dot{q}_1},\frac{\partial L}{\partial \dot{q}_2},\cdots \frac{\partial L}{\partial \dot{q}_N}\right) = (p_1, p_2\cdots p_N)\,,</math>

and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

:<math>H(\mathbf{q},\mathbf{p},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t)</math>

where <math>\cdot</math> denotes the [[dot product]], also leading to [[Hamiltonian mechanics|Hamilton's equations]]:

:<math>\mathbf{\dot{p}} = - \frac{\partial H}{\partial \mathbf{q}}\,,\quad \mathbf{\dot{q}} = + \frac{\partial H}{\partial \mathbf{p}} \,,</math>

which are now a set of 2''N'' first-order ordinary differential equations, one for each ''q<sub>i</sub>''(''t'') and ''p<sub>i</sub>''(''t''). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

:<math>\frac{dH}{dt}=-\frac{\partial L}{\partial t}\,,</math>

which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:

:<math>\mathbf{\dot{p}} = \boldsymbol{\mathcal{Q}}\,.</math>

Analogous to the configuration space, the set of all momenta is the '''generalized [[momentum space]]''':

:<math>\mathcal{M} = \{ \mathbf{p}\in\mathbb{R}^N \}\,.</math>

("Momentum space" also refers to "'''k'''-space"; the set of all [[wave vector]]s (given by [[De Broglie relation]]s) as used in quantum mechanics and theory of [[wave]]s)

The set of all positions and momenta form the '''[[phase space]]''':

:<math>\mathcal{P} = \mathcal{C}\times\mathcal{M} = \{ (\mathbf{q},\mathbf{p})\in\mathbb{R}^{2N} \} \,,</math>

that is, the [[Cartesian product]] of the configuration space and generalized momentum space.

A particular solution to Hamilton's equations is called a ''[[Phase portrait|phase path]]'', a particular curve ('''q'''(''t''),'''p'''(''t'')) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the ''[[phase portrait]]'':

:<math>\{ (\mathbf{q}(t),\mathbf{p}(t))\in\mathbb{R}^{2N}\,:\,t\ge0, t\in\mathbb{R} \} \subseteq \mathcal{P}\,,</math>

===The Poisson bracket===
{{main |Poisson bracket}}

All dynamical variables can be derived from position '''q''', momentum '''p''', and time ''t'', and written as a function of these: ''A'' = ''A''('''q''', '''p''', ''t''). If ''A''('''q''', '''p''', ''t'') and ''B''('''q''', '''p''', ''t'') are two scalar valued dynamical variables, the ''Poisson bracket'' is defined by the generalized coordinates and momenta:

:<math>
\begin{align}
\{A,B\}  \equiv \{A,B\}_{\mathbf{q},\mathbf{p}} & = \frac{\partial A}{\partial \mathbf{q}}\cdot\frac{\partial B}{\partial \mathbf{p}} - \frac{\partial A}{\partial \mathbf{p}}\cdot\frac{\partial B}{\partial \mathbf{q}}\\
& \equiv \sum_k \frac{\partial A}{\partial q_k}\frac{\partial B}{\partial p_k} - \frac{\partial A}{\partial p_k}\frac{\partial B}{\partial q_k}\,,
\end{align}</math>

Calculating the [[total derivative]] of one of these, say ''A'', and substituting Hamilton's equations into the result leads to the time evolution of ''A'':

:<math> \frac{dA}{dt} = \{A,H\} + \frac{\partial A}{\partial t}\,. </math>

This equation in ''A'' is closely related to the equation of motion in the [[Heisenberg picture]] of [[quantum mechanics]], in which classical dynamical variables become [[operator (physics)|quantum operators]] (indicated by hats (^)), and the Poisson bracket is replaced by the [[commutator]] of operators via Dirac's [[canonical quantization]]:

:<math>\{A,B\} \rightarrow \frac{1}{i\hbar}[\hat{A},\hat{B}]\,.</math>