Conservative force is gradient of potential energy
==================================================

A conservative force is a such a force, the total work of which in moving a particle between two
points is independent of the path taken. Alternative definition states that if a particle travels
in a closed loop, the total work done by a conservative force is zero.

**Conditions:**

#. Force is conservative. Mathematically, this can be expressed as
   :math:`\text{curl} \, {\vec F} \! \left( \vec r \right) \equiv 0`, i.e. the force field must
   be irrotational.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Conservative_force>`__.

.. py:currentmodule:: symplyphysics.laws.dynamics.fields.conservative_force_is_gradient_of_potential_energy

.. py:data:: position_vector

    Position vector of a point in space. See :attr:`~symplyphysics.symbols.classical_mechanics.distance_to_origin`.

Symbol:
    :code:`r`

Latex:
    :math:`{\vec r}`

Dimension:
    :code:`length`


.. py:data:: force

    Vector field of the conservative force as a function of the :attr`position_vector`. See
    :attr:`~symplyphysics.symbols.classical_mechanics.force`.

Symbol:
    :code:`F(r)`

Latex:
    :math:`{\vec F} \left( {\vec r} \right)`

Dimension:
    :code:`force`


.. py:data:: potential_energy

    Scalar field of the force's potential as a function of the :attr`position_vector`. See
    :attr:`~symplyphysics.symbols.classical_mechanics.potential_energy`.

Symbol:
    :code:`U(r)`

Latex:
    :math:`U{\left({\vec r} \right)}`

Dimension:
    :code:`energy`


.. py:data:: law

    :code:`F(r) = -grad(U(r))`


    Latex:
        .. math::
            {\vec F} \left( {\vec r} \right) = - \text{grad} \, U{\left({\vec r} \right)}