Conservative Force

A force is known as conservative when the work produces by this force is independent of the way followed by its point of action. If it is not the case it then known as not-conservative .

This type of force has three remarkable properties:

1. the mechanical energy of a system only subjected to the action of conservative forces is preserved.
2. There exists a scalar Champ $U\; \backslash ,$, also called Potentiel, such as the force is written $\backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash ,\; U\; \backslash ,$.
3. If the point of application of a conservative force moves of a point has at a point B, work $W\; \backslash ,$ of the force in question is obtained simply starting from the potential $U\; \backslash ,$: $W=U\; (A)\; -\; U\; (B)\; \backslash ,$.

Current examples of conservative and not-conservative forces

• the electric force which derives from the electric Potentiel.
• the gravitational Force which derives from the Potentiel of gravitation.
• the Force of Lorentz not working but not deriving from a potential, one thus does not place them in the category of the conservative forces.
• forces of Friction, which it is about solid Frottement or of fluid Frottement is not conservative forces because their work depends explicitly on the way followed by the system.
• the forces of Pression are not conservative because during the evolution of energy is transmitted to the environment external with the system.

Independence of the followed way

Let us consider a material particle moving of a point has towards a point B, and on which is exerted a conservative force $\backslash \; vec\; \{F\}$, then the work produced by this force does not depend on the way followed by the solid. Thus for two trajectories C₁ and C₂ connecting the point has at the point B, the force provides same work:

$W=\; \backslash \; int\_\; \{C\_1\}\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}\; =\; \backslash \; int\_\; \{C\_2\}\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}$

An immediate consequence of this property is that in the case of a closed trajectory (if the particle turns over to its initial position), the work of a conservative force is null.

Potential of a conservative force

Existence of the potential

Let us consider a conservative force now function of the position of its point of application, i.e. such as $\backslash \; vec\; \{F\}$ is a function of the coordinates $x\; \backslash ,$, $y\; \backslash ,$ and $z\; \backslash ,$ then, under the terms of the independence of the followed way, whatever the closed trajectory $\backslash \; mathcal\; \{C\}\; \backslash ,$, the work of the force is null: $W=\; \backslash \; oint\_\; \{\backslash \; mathcal\; \{C\}\}\; \backslash \; vec\; \{F\}\; (X,\; there,\; Z)\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}\; =0$ which one deduces according to the Théorème from Stokes that $\backslash \; vec\; \{\backslash \; nabla\}\; \backslash \; wedge\; \backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{0\}$. This last relation implies the existence of a scalar field $U\; (X,\; there,\; Z)\; \backslash ,$ such as: $\backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash ,\; U=-\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash ,\; U$ CQFD.

The field $U\; \backslash ,$ is called potential force and is homogeneous with a energy. From its definition, the field $U\; \backslash ,$ is defined except for a constant. The value of the latter is generally arbitrary, in which case it is selected in order to simplify calculations.

Examples of fields are given in the article on the Potentiel.

Reciprocal

Reciprocally, let us consider a force $\backslash \; vec\; \{F\}$ drifting of a potential $U\; \backslash ,$: $\backslash \; vec\; \{F\}\; =\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash ,\; U=-\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; \backslash ,\; U$ By noticing that $dU=\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash ,\; U\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}$ is a total differential , one finds that the work of the force takes the following expression:

$W=\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}\; =\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; -\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash ,\; U\; \backslash \; cdot\; \backslash \; vec\; \{DLL\}$

- \ int_ {has} ^ {B} dUU (A) - U (B)

Work thus does not depend that value of the potential at the points has and B. the work of a force deriving from a potential thus does not depend on the followed way, such a force is thus conservative . CQFD.

Mechanical conservation of energy

The conservative forces are thus called parce the mechanical energy of a system subjected to the action of conservative forces is constant: the energy of the system is preserved.

This property is an immediate consequence of the theorem of the kinetic energy. For a solid traversing a trajectory connecting a point has at a point B and subjected to a conservative force of potential $U\; \backslash ,$, there is on the one hand equality between the variation of the kinetic energy and the work of the force:

$E\_c\; (B)\; -\; E\_c\; (A)=W\; \backslash ,$

and in addition, the work of the conservative force which is obtained starting from the variation of the potential between the points has and b:

$W=U\; (A)\; -\; U\; (B)\; \backslash ,$

from which one deduces the following equality immediately:

$E\_c\; (B)\; +\; U\; (B)=E\_c\; (A)\; +\; U\; (A)\; \backslash ,$

It is thus seen that the sum of the kinetic energy and the potential is preserved. This quantity is precisely the mechanical energy system. The expression above shows clearly that total energy is distributed between the kinetic energy and the potential, and can thus pass successively from the one to the other. This is why the potential $U\; \backslash ,$ is also called potential energy: it is energy which can be potentially transformed into kinetic energy.

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