# ⓘ Conceptual programs in physics. For the simple case of single particle with mass m moving along one dimension x and acted upon by forces F i {\displaystyle F_{i ..

## ⓘ Conceptual programs in physics

For the simple case of single particle with mass m moving along one dimension x and acted upon by forces F i {\displaystyle F_{i}}, the program of classical mechanics is to determine the state x: R → R {\displaystyle x:\mathbb {R} \to \mathbb {R} } by solving Newtons second law,

In Lagrangian mechanics for the same system, the state q: R → R 3 N {\the style property display set to \mathbf {U}:\mathbb {R} \to \mathbb {R} ^{3P}} solves Hamiltons principle δ S = 0 {\properties display style set to \Delta S=0} where the action functional is defined how

S \ {\stackrel {\mathrm {def} }{=}}\int _{t_{1}}^{t_{2}}L\mathbf {q} t,{\dot {\mathbf {q} }}t,t)\,\mathrm {d} t}.

In Hamiltonian mechanics with canonical coordinates q, p {\the style property display set to \mathbf {m},\mathbf {p}} and Hamiltonian function H q, p, t {\the style property display the value of {\mathcal {h}}\mathbf {m},\mathbf {p},t}, the state q: R → R 3 N {\the style property display set to \mathbf {U}:\mathbb {R} \to \mathbb {R} ^{3P}} is determined by solving

q ′ t = ∂ H ∂ p, p ′ t = − ∂ H ∂ q, ∂ H ∂ t = ∂ L ∂ t {\displaystyle \mathbf {q} t={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p} }}\quad,\quad \mathbf {p} t=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {q} }}\quad,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}={\frac {\partial {\mathcal {L}}}{\partial t}}}.

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