reperiendi

Imaginary time

Posted in General physics by Mike Stay on 2009 January 9
Statics (geometric = no time):
\displaystyle x [x] x coordinate
\displaystyle y(x) [y] y coordinate
\displaystyle k [k] proportionality constant
\displaystyle y'(x) = \frac{dy(x)}{dx} [y/x] slope
\displaystyle s(x) = ky'(x) [k y/x] proportional to slope
\displaystyle T(x) = \frac{1}{2} ky'(x)^2 [k y^2/x^2] distortion
\displaystyle V(y(x)) [k y^2/x^2] original shape
\displaystyle S(y) = \int (T + V\circ y)(x) dx
\displaystyle = \int \left[ \frac{k}{2} \left(\frac{dy(x)}{dx}\right)^2 + V(y(x)) \right] dx
[k y^2/x] least S at equilibrium
Statics (with energy):
\displaystyle x [x] parameterization of curve
\displaystyle y(x) [y] y coordinate
\displaystyle k [kg x/s^2] spring constant at x
\displaystyle v(x) = \frac{dy(x)}{dx} [y/x] slope
\displaystyle F(x) = kv(x) [kg y/s^2] force due to stretching
\displaystyle T(x) = \frac{1}{2} kv(x)^2 [kg y^2/s^2 x = J/x] stretching energy density
\displaystyle V(y(x)) [kg y^2/s^2 x= J/x] gravitational energy density
\displaystyle E(y) = \int (T + V\circ y)(x) dx
\displaystyle = \int \left[ \frac{k}{2} \left(\frac{dy(x)}{dx}\right)^2 + V(y(x)) \right] dx
[kg y^2/s^2 = J] energy (least energy at equilibrium)
Statics (unitless distance):
\displaystyle x [1] parameterization of curve
\displaystyle y(x) [m] y coordinate
\displaystyle k [kg/s^2] spring constant
\displaystyle v(x) = \frac{dy(x)}{dx} [m] relative displacement
\displaystyle F(x) = kv(x) [kg m/s^2 = N] force at x due to stretching
\displaystyle T(x) = \frac{1}{2} kv(x)^2 [kg m^2 / s^2 = J] stretching energy at x
\displaystyle V(y(x)) [kg m^2 / s^2 = J] gravitational energy at x
\displaystyle E(y) = \int (T + V\circ y)(x) dx
\displaystyle = \int \left[ \frac{k}{2} \left(\frac{dy(x)}{dx}\right)^2 + V(y(x)) \right] dx
[kg m^2 / s^2 = J] energy (least energy at equilibrium)
Dynamics (\displaystyle \underline{\lambda x.y(x) \mapsto \lambda t.y(it)}):
\displaystyle t [s] time
\displaystyle y(it) [m] y coordinate
\displaystyle m [kg] mass
\displaystyle v(t) = \frac{dy(it)}{dt} = i\frac{dy(it)}{dit} [m/s] velocity
\displaystyle p(t) = m v(t) [kg m/s] momentum
\displaystyle T(t) = \frac{1}{2}m\;v(t)^2 = -\frac{m}{2}\left(\frac{dy(it)}{dit}\right)^2 [-kg m^2/s^2 = -J] -kinetic energy
\displaystyle V(y(it)) [kg m^2 / s^2 = J] potential energy
\displaystyle iS(y) = \int (T + V\circ y \circ i)(t) dt
\displaystyle = \int \left[ -\frac{m}{2}\left(\frac{dy(it)}{dit}\right)^2 + V(y(it)) \right] dt
\displaystyle = i \int \left[ \frac{m}{2}\left(\frac{dy(it)}{dit}\right)^2 - V(y(it)) \right] d it

[kg m^2/s] i * action

See also Toby Bartels‘ sci.physics post.

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One Response

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  1. partandwhole said, on 2009 January 12 at 1:31 pm

    I am fascinated. And rueful over my failed phone call this weekend. And massively sleep-deprived after a full-weekend battle with children-borne night terrors and consequent lack of sleep. So: when I call you up (tonight?) can we please talk also about material manifestations, if such there may be, of imaginary time? Does the universe actually manifest this and its manifold derivatives?


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