Three Dimensional Forces

Mathematics for Physics 3: Dynamics and Differential Equations

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Lecture 17: Conservative forces in three dimensions

So far we discussed conservative forces in one dimension, defining them as ones that can be obtained from a derivative of some potential V (x). A conservative force acting along the x axis is a force that can be written as: ˆ dV (x) . dx More generally, one can define conservative forces in three dimensions by introducing a potential function that depends on all three coordinates V (x, y, z). We then have: Fx = − Fx = − ∂V (x, y, z) , ∂x Fy = − ∂V (x, y, z) , ∂y Fz = − ∂V (x, y, z) , ∂z (285)

where we denoted partial differentiation with respect to a given cordinate by ∂. This can be written in a vector form as follows: F = Fx x + Fy y + Fz z = − ˆ ˆ ˆ ∂V (x, y, z) ∂V (x, y, z) ∂V (x, y, z) x− ˆ y− ˆ z ≡ −∇V (x, y, z) , ˆ ∂x ∂y ∂z (286)

where ∇, called the gradient, is a vector differential operator: it acts on a scalar function of the coordinates V (x, y, z) and produces a vector. Each component of the vector is the slope in the respective direction. The direction of ∇V (x, y, z) at a given point (x, y, z) is the direction along which the local change of the function V (x, y, z) is maximal. The simplest example is that of Galilean gravity where V (x, y, z) = mgz, where z is the vertical coordinate. The gravitational potential increases as z increases, and it does not depend at all on the horizontal coordinates x and y. The corresponding force is: ￿ ￿ ∂mgz ∂mgz ∂mgz F = −∇V (x, y, z) = −∇mgz = − x+ ˆ y+ ˆ z = −mg [0ˆ + 0ˆ + 1ˆ] = −mgˆ ˆ x y z z ∂x ∂y ∂z which is indeed in the direction −ˆ the direction along which the potential changes. z It is often convenient to denote the set of coordinates (x, y, z) by the vector r. We can think of the potential as a function of the coordinate vector r. We defined a conservative force as one that can be written as a derivative of a scalar function of the coordinates: F = −∇V (r) . (287) Alternative (and...