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The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
List of textbooks in physics: Category:Physics textbooks; List of textbooks on classical mechanics and quantum mechanics; List of textbooks in electromagnetism; List of textbooks on relativity; List of textbooks in thermodynamics and statistical mechanics
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [8] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependence, i.e., = (,) and = (,).The functional form of Hamilton's equations is ˙ =, ˙ = In general, a transformation (q, p) → (Q, P) does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some ...
The formulas given above for T, N, and B depend on the curve being given in terms of the arclength parameter. This is a natural assumption in Euclidean geometry, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of gauge. However, it may be awkward to ...
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal, one-to-one transformations on coordinate space-time. [ 1 ] [ 2 ] They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically transformed by ...
Often, theory can establish the existence of a change of variables, although the formula itself cannot be explicitly stated. For an integrable Hamiltonian system of dimension n {\displaystyle n} , with x ˙ i = ∂ H / ∂ p j {\displaystyle {\dot {x}}_{i}=\partial H/\partial p_{j}} and p ˙ j = − ∂ H / ∂ x j {\displaystyle {\dot {p}}_{j ...
A Magic Triangle image mnemonic - when the terms of Ohm's law are arranged in this configuration, covering the unknown gives the formula in terms of the remaining parameters. It can be adapted to similar equations e.g. F = ma, v = fλ, E = mcΔT, V = π r 2 h and τ = rF sinθ.