Time derivative

The time derivative is a derivative of a value with respect to time . From the location of a body , the speed , the acceleration and the jerk are generated through time derivative applied several times in a row . In general, the rate of change of the value arises through time derivative , which, as in the case of location, can be a physical quantity or, for example, an economic function.

The reversal of the time derivative is the time integration , for in the form of numerical simulation are powerful solution methods. In this way, predictions can be made about future values, which help with evaluation and / or decision-making . In particular, the assumptions and theories behind the time derivative can be validated or falsified . In the philosophy of science according to Karl Popper , the falsifiability of a theory or hypothesis plays a central role.

Usually from Latin tempus is the variable that denotes time. ${\displaystyle t}$

notation

→ Main article : differential calculus

Many notations are used for the time derivative of a function ( ). The Leibniz notation is based on Gottfried Wilhelm Leibniz ${\displaystyle f}$

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}}$

back. Isaac Newton used the superpoint (Newton notation)

${\displaystyle {\dot {f}}}$

which is often used in physics. If the function value depends not only on the time, but also on other quantities , then the partial derivative means ${\displaystyle x}$

${\displaystyle {\frac {\partial f(x,t)}{\partial t}}:=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}f(x,t)\right|_{x\;{\text{fest}}}}$

the time derivative when held constant . Multiple time derivatives, such as the second, are noted as ${\displaystyle x}$

${\displaystyle {\frac {\mathrm {d} ^{2}f}{\mathrm {d} t^{2}}},\;{\ddot {f}},\;{\frac {\partial ^{2}f}{\partial t^{2}}}}$

Time derivatives for vector quantities are written in the same way :

${\displaystyle {\vec {v}}={\begin{pmatrix}v_{1}&v_{2}&v_{3}&\ldots \end{pmatrix}}\;\rightarrow {\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}={\begin{pmatrix}{\frac {\mathrm {d} v_{1}}{\mathrm {d} t}}&{\frac {\mathrm {d} v_{2}}{\mathrm {d} t}}&{\frac {\mathrm {d} v_{3}}{\mathrm {d} t}}&&\ldots \end{pmatrix}}}$

In order to be able to carry out the derivations at all, the time is assumed to be a continuous variable. This assumption is discussed in the article "Time", see there " Limits of the physical concept of time ".

Special time derivatives

Relative time derivative

→ Main article : Accelerated frame of reference

On earth, everyday speeds are measured relative to the environment. For example, the speedometer in the car measures the speed relative to the ground. In addition , however, the earth rotates on itself . If this is to be taken into account, then a portion is added to the former local or relative speed on the earth's surface, which results from the rotation of the earth:

${\displaystyle {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}={\dot {\vec {x}}}_{\text{rel}}+{\dot {\vec {x}}}_{\text{rot}}}$

Mathematically, this can be represented with a rotating reference system.

Let be a vector with components with respect to a basis system . According to the product rule , the time derivative is: ${\displaystyle \textstyle {\vec {x}}=\sum _{i}x_{i}{\hat {e}}_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle \{{\hat {e}}_{i}\}}$

${\displaystyle {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}=\underbrace {\sum _{i}{\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}{\hat {e}}_{i}} _{{\dot {\vec {x}}}_{\text{rel}}}+\underbrace {\sum _{i}x_{i}{\frac {\mathrm {d} {\hat {e}}_{i}}{\mathrm {d} t}}} _{{\dot {\vec {x}}}_{\text{rot}}}}$

In it is

${\displaystyle {\dot {\vec {x}}}_{\text{rel}}={\frac {\mathrm {d} '{\vec {x}}}{\mathrm {d} t}}={\frac {\mathrm {d} _{r}{\vec {x}}}{\mathrm {d} t}}}$

the relative time derivative to the base system , where this is assumed to be constant. ${\displaystyle \{{\hat {e}}_{i}\}}$

In the case of an orthonormal system , only a rotation of the reference system comes into question, in which the time derivative of the basis vectors in three-dimensional space is calculated from the cross product with the angular velocity of the reference system. This results in ${\displaystyle {\dot {\hat {e}}}_{i}={\vec {\omega }}\times {\hat {e}}_{i}}$ ${\displaystyle \times }$ ${\displaystyle {\vec {\omega }}}$

${\displaystyle {\dot {\vec {x}}}_{\text{rot}}={\vec {\omega }}\times {\vec {x}}}$

and the complete time derivative

${\displaystyle {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}={\frac {\mathrm {d} '{\vec {x}}}{\mathrm {d} t}}+{\vec {\omega }}\times {\vec {x}}}$

Local and material time derivative

In the case of an extended body, a variable assigned to it, for example the temperature , in the case of uneven distribution , can depend on the location or on the particle of the body being considered . The time derivative of such a quantity can be evaluated accordingly: ${\displaystyle T}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\mathcal {P}}}$

• with a fixed point in space ( local time derivative ) or
• with a captured particle ( material or substantial derivative ).

Because the physical laws in classical mechanics refer to material points, the substantial time derivative is decisive there.

The partial derivative of time

${\displaystyle {\frac {\partial T({\vec {x}},t)}{\partial t}}=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}T({\vec {x}},t)\right|_{{\vec {x}}\;{\text{fest}}}}$

is the local time derivative, i.e. H. the rate of change that can be observed at a fixed point in space . For example, an outdoor thermometer measures the temperature where it is attached. ${\displaystyle {\vec {x}}}$

The material time derivative is the time derivative with the particle held . The thermometer would only measure the temperature and its rate at the particle . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$

In Lagrange's representation , the material time derivative is the partial derivative with respect to time:

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}T_{0}({\mathcal {P}},t):=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}T_{0}({\mathcal {P}},t)\right|_{{\mathcal {P}}\;{\text{fest}}}={\frac {\partial }{\partial t}}T_{0}({\mathcal {P}},t)}$

In Euler's representation , the material time derivative is made up of the local and an additional convective part:

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}T({\vec {x}},t)={\frac {\partial T({\vec {x}},t)}{\partial t}}+\operatorname {grad} {\bigl (}T({\vec {x}},t){\bigr )}\cdot {\vec {v}}({\vec {x}},t)}$

With

• the temperature gradient ${\displaystyle \operatorname {grad} {\bigl (}T({\vec {x}},t){\bigr )}={\frac {\partial T({\vec {x}},t)}{\partial {\vec {x}}}}}$
• the velocity of the particle currently in place . ${\displaystyle {\vec {v}}({\vec {x}},t)={\frac {\partial {\vec {x}}}{\partial t}}}$ ${\displaystyle t}$ ${\displaystyle {\vec {x}}}$

See also: Different usage of the terms partial and total derivative in physics

Objective time derivative

→ Main article : Euclidean transformation

An occupant of a moving train will judge the speed of a bird flying past differently than a pedestrian in the vicinity. The speed is therefore dependent on the point of view; more precisely, it is not invariant to the reference system or, for a shorter time, not objective .

For the formulation of a material model in which the rates of constitutive variables occur, as for example in the case of Newtonian fluid , objective time derivatives of these variables are required. Because it does not correspond to the experience that a moving observer measures a different material behavior than a stationary one.

For an objective spatial vector field , for example, the time derivative is again objective; therein is the velocity gradient . ${\displaystyle {\vec {y}}({\vec {x}},t)}$ ${\displaystyle {\stackrel {\nabla }{\vec {y}}}={\dot {\vec {y}}}-\mathbf {l} \cdot {\vec {y}}}$ ${\displaystyle \mathbf {l} }$

Particularly elegant formulations for objective time derivatives result in convective coordinates .

use

physics

Time derivatives are a key term in physics , where they appear in many basic equations, including:

• The force is the time derivative of the momentum ,
• The power is the time derivative of the energy and
• the electric current is the time derivative of the electric charge .

chemistry

The theory of the transition state enables the determination of the absolute reaction rate constants of a chemical reaction . This reaction rate can then be used in a rate equation , which is a first order differential equation in time.

biology

The population dynamics is the change in the size of biological populations in shorter or longer periods. The time derivative of the population size is the difference between the birth rate and the death rate, which in turn are influenced by the population size.

Economics

In economics , theoretical models, for example the Solow model , describe the behavior of economic variables over time. Time derivatives of the economic variables occur:

• For stock investment include changes in inventories of stocks and supplies , so their time derivatives.
• The speed of circulation of money is the quotient of the nominal gross domestic product and the money supply itself.

literature

• Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 978-3-642-62463-6 , pp. 4 , doi : 10.1007 / 978-3-642-55485-8 ( limited preview in Google Book search).
• Holm Altenbach : Continuum Mechanics . Introduction to the material-independent and material-dependent equations. Springer-Verlag, Berlin, Heidelberg 2012, ISBN 978-3-642-24118-5 , pp. 230 ff ., doi : 10.1007 / 978-3-642-24119-2 .

Web links

• Johannes Strommer: Distance, speed, acceleration, jolt. November 8, 2018, accessed May 1, 2019 . 

Individual evidence

1. ↑ Mathematics for Economists: Economic Functions. Wikibooks, accessed November 27, 2018 . 
2. ↑ see for example
   • Nürnberger, D .: Implicit time integration for the simulation of turbomachine flows. (html) DLR, Library and Information System, 2004, accessed on May 12, 2019 . 
   • Marcus Wagner: Linear and non-linear FEM . Springer-Verlag, Regensburg 2017, ISBN 978-3-658-17865-9 , time integration of nonlinear dynamic problems, p. 223-246 , doi : 10.1007 / 978-3-658-17866-6 .
   • Kuhl, D .: Robust Time Integration in Nonlinear Elastodynamics . Ed .: Institute for Construction Mechanics and Numerical Mechanics, University of Hanover. Hanover 1996 ( dlr.de [accessed on May 12, 2019]).
3. ^ Rolf Mahnken: Textbook of technical mechanics . Dynamics: A vivid introduction. Springer-Verlag, Heidelberg, Dordrecht, London, New York 2011, ISBN 978-3-642-19837-3 , pp. 282 ff ., doi : 10.1007 / 978-3-642-19838-0 ( limited preview in Google Book Search [accessed on November 27, 2018]).
4. ↑ Greve (2003), p. 3 f., Altenbach (2012), p. 76 ff.
5. ↑ Greve (2003), p. 42 ff., Altenbach (2012), p. 230 ff.