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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Effective potential</span></span>
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<p>The <b>effective potential</b> (also known as <b>effective potential energy</b>) combines multiple, perhaps opposing, effects into a single <a href="Potential" title="Potential">potential</a>. In its basic form, it is the sum of the "opposing" <a href="Centrifugal_force" title="Centrifugal force">centrifugal</a> potential energy with the <a href="Potential_energy" title="Potential energy">potential energy</a> of a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. It may be used to determine the <a href="Orbit" title="Orbit">orbits</a> of planets (both <a href="Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian</a> and <a href="General_relativity" title="General relativity">relativistic</a>) and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer <a href="Dimension" title="Dimension">dimensions</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>The basic form of potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}}</annotation>
</semantics>
</math></span><img src="./dfda110e5eef417df92c199da60b20b21ef5e639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.557ex; height:2.509ex;" alt="{\displaystyle U_{\text{eff}}}" loading="lazy"></span> is defined as
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}(\mathbf {r} )={\frac {L^{2}}{2\mu r^{2}}}+U(\mathbf {r} ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<mi>μ<!-- μ --></mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}(\mathbf {r} )={\frac {L^{2}}{2\mu r^{2}}}+U(\mathbf {r} ),}</annotation>
</semantics>
</math></span></span>
where
</p>
<dl><dd><i>L</i> is the <a href="Angular_momentum" title="Angular momentum">angular momentum</a>,</dd>
<dd><i>r</i> is the distance between the two masses,</dd>
<dd><i>μ</i> is the <a href="Reduced_mass" title="Reduced mass">reduced mass</a> of the two bodies (approximately equal to the mass of the orbiting body if one mass is much larger than the other),</dd>
<dd><i>U</i>(<i>r</i>) is the general form of the <a href="Potential" title="Potential">potential</a>.</dd></dl>
<p>The effective force, then, is the negative <a href="Gradient" title="Gradient">gradient</a> of the effective potential:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {F} _{\text{eff}}&=-\nabla U_{\text{eff}}(\mathbf {r} )\\&={\frac {L^{2}}{\mu r^{3}}}{\hat {\mathbf {r} }}-\nabla U(\mathbf {r} ),\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
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<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi>μ<!-- μ --></mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>,</mo>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {F} _{\text{eff}}&=-\nabla U_{\text{eff}}(\mathbf {r} )\\&={\frac {L^{2}}{\mu r^{3}}}{\hat {\mathbf {r} }}-\nabla U(\mathbf {r} ),\end{aligned}}}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation>
</semantics>
</math></span><img src="./9740464b71653e12932278ee944540be8caa5b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}" loading="lazy"></span> denotes a unit vector in the radial direction.
</p>
<div class="mw-heading mw-heading2"><h2 id="Important_properties">Important properties</h2></div>
<p>There are many useful features of the effective potential, such as
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}\leq E.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>E</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}\leq E.}</annotation>
</semantics>
</math></span></span>
</p><p>To find the radius of a circular orbit, simply minimize the effective potential with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, or equivalently set the net force to zero and then solve for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation>
</semantics>
</math></span><img src="./fb12fcfddb65e3d1e6a044215f6e833f0cd4337b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}" loading="lazy"></span>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU_{\text{eff}}}{dr}}=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>r</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {dU_{\text{eff}}}{dr}}=0.}</annotation>
</semantics>
</math></span></span>
After solving for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation>
</semantics>
</math></span><img src="./fb12fcfddb65e3d1e6a044215f6e833f0cd4337b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}" loading="lazy"></span>, plug this back into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}}</annotation>
</semantics>
</math></span><img src="./dfda110e5eef417df92c199da60b20b21ef5e639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.557ex; height:2.509ex;" alt="{\displaystyle U_{\text{eff}}}" loading="lazy"></span> to find the maximum value of the effective potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}^{\text{max}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mtext>max</mtext>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}^{\text{max}}}</annotation>
</semantics>
</math></span><img src="./2f1b5f58d0e5c5944d84054e71b5ea3c0fbfe927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.132ex; height:2.843ex;" alt="{\displaystyle U_{\text{eff}}^{\text{max}}}" loading="lazy"></span>.
</p><p>A circular orbit may be either stable or unstable. If it is unstable, a small perturbation could destabilize the orbit, but a stable orbit would return to equilibrium. To determine the stability of a circular orbit, determine the concavity of the effective potential. If the concavity is positive,
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}U_{\text{eff}}}{dr^{2}}}>0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}U_{\text{eff}}}{dr^{2}}}>0,}</annotation>
</semantics>
</math></span></span>
the orbit is stable.
</p><p>The frequency of small oscillations, using basic <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> analysis, is
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {\frac {U_{\text{eff}}''}{m}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<msubsup>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
<mo>″</mo>
</msubsup>
<mi>m</mi>
</mfrac>
</msqrt>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\frac {U_{\text{eff}}''}{m}}},}</annotation>
</semantics>
</math></span></span>
where the double prime indicates the second derivative of the effective potential with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> and is evaluated at a minimum.
</p>
<div class="mw-heading mw-heading2"><h2 id="Gravitational_potential">Gravitational potential</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gravitational_potential" title="Gravitational potential">Gravitational potential</a></div>
<p>Consider a particle of mass <i>m</i> orbiting a much heavier object of mass <i>M</i>. Assume <a href="Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, which is both classical and non-relativistic. The conservation of <a href="Energy" title="Energy">energy</a> and <a href="Angular_momentum" title="Angular momentum">angular momentum</a> give two constants <i>E</i> and <i>L</i>, which have values
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
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<mo>)</mo>
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<mo>−<!-- − --></mo>
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<mfrac>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},}</annotation>
</semantics>
</math></span></span>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=mr^{2}{\dot {\phi }},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
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<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle L=mr^{2}{\dot {\phi }},}</annotation>
</semantics>
</math></span></span>
when the motion of the larger mass is negligible. In these expressions,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {r}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>r</mi>
<mo>˙<!-- ˙ --></mo>
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<annotation encoding="application/x-tex">{\displaystyle {\dot {r}}}</annotation>
</semantics>
</math></span><img src="./99b51a39764cf6daa615ef2144ac420ecff01cfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\dot {r}}}" loading="lazy"></span> is the derivative of <i>r</i> with respect to time,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\phi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\dot {\phi }}}</annotation>
</semantics>
</math></span><img src="./0446aa46e762e6b105ed6cd084731c4a37b8a3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.467ex; height:3.009ex;" alt="{\displaystyle {\dot {\phi }}}" loading="lazy"></span> is the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> of mass <i>m</i>,</dd>
<dd><i>G</i> is the <a href="Gravitational_constant" title="Gravitational constant">gravitational constant</a>,</dd>
<dd><i>E</i> is the total energy,</dd>
<dd><i>L</i> is the <a href="Angular_momentum" title="Angular momentum">angular momentum</a>.</dd></dl>
<p>Only two variables are needed, since the motion occurs in a plane. Substituting the second expression into the first and rearranging gives
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\dot {r}}^{2}=2E-{\frac {L^{2}}{mr^{2}}}+{\frac {2GmM}{r}}=2E-{\frac {1}{r^{2}}}\left({\frac {L^{2}}{m}}-2GmMr\right),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>r</mi>
<mo>˙<!-- ˙ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>=</mo>
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<mi>E</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msup>
<mrow>
<mi>m</mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msup>
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</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mi>G</mi>
<mi>m</mi>
<mi>M</mi>
</mrow>
<mi>r</mi>
</mfrac>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>E</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>G</mi>
<mi>m</mi>
<mi>M</mi>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m{\dot {r}}^{2}=2E-{\frac {L^{2}}{mr^{2}}}+{\frac {2GmM}{r}}=2E-{\frac {1}{r^{2}}}\left({\frac {L^{2}}{m}}-2GmMr\right),}</annotation>
</semantics>
</math></span></span>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}m{\dot {r}}^{2}=E-U_{\text{eff}}(r),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>m</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mi>E</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
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<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}m{\dot {r}}^{2}=E-U_{\text{eff}}(r),}</annotation>
</semantics>
</math></span></span>
where
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{eff}}(r)={\frac {L^{2}}{2mr^{2}}}-{\frac {GmM}{r}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>eff</mtext>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<mi>m</mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>G</mi>
<mi>m</mi>
<mi>M</mi>
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<mi>r</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{\text{eff}}(r)={\frac {L^{2}}{2mr^{2}}}-{\frac {GmM}{r}}}</annotation>
</semantics>
</math></span></span>
is the effective potential.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> The original two-variable problem has been reduced to a one-variable problem. For many applications the effective potential can be treated exactly like the potential energy of a one-dimensional system: for instance, an energy diagram using the effective potential determines turning points and locations of stable and unstable <a href="Mechanical_equilibrium" title="Mechanical equilibrium">equilibria</a>. A similar method may be used in other applications, for instance, determining orbits in a general relativistic <a href="Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a>.
</p><p>Effective potentials are widely used in various condensed matter subfields, e.g. the Gauss-core potential (Likos 2002, Baeurle 2004) and the screened <a href="Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a> (Likos 2001).
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Geopotential" title="Geopotential">Geopotential</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">A similar derivation may be found in José & Saletan, <i>Classical Dynamics: A Contemporary Approach</i>, pp. 31–33.</span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFSeidov2004" class="citation journal cs1">Seidov, Zakir F. (2004). "The Roche Problem: Some Analytics". <i>The Astrophysical Journal</i>. <b>603</b>: <span class="nowrap">283–</span>284. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/astro-ph/0311272">astro-ph/0311272</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004ApJ...603..283S">2004ApJ...603..283S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F381315">10.1086/381315</a>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><cite id="CITEREFJoséSaletan1998" class="citation book cs1">José, J. V.; Saletan, E. J. (1998). <i>Classical Dynamics: A Contemporary Approach</i> (1st ed.). Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-63636-0</bdi>.</cite>.</li>
<li><cite id="CITEREFLikosRosenfeldtDingenoutsBallauff2002" class="citation journal cs1">Likos, C. N.; Rosenfeldt, S.; Dingenouts, N.; <a href="Matthias_Ballauff" title="Matthias Ballauff">Ballauff, M.</a>; Lindner, P.; Werner, N.; Vögtle, F.; et al. (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110719010918/http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no">"Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study"</a>. <i>J. Chem. Phys</i>. <b>117</b> (4): <span class="nowrap">1869–</span>1877. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JChPh.117.1869L">2002JChPh.117.1869L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1486209">10.1063/1.1486209</a>. Archived from <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no">the original</a></span> on 2011-07-19.</cite></li>
<li><cite id="CITEREFBaeurleKroener_J.2004" class="citation journal cs1">Baeurle, S. A.; Kroener J. (2004). "Modeling Effective Interactions of Micellar Aggregates of Ionic Surfactants with the Gauss-Core Potential". <i>J. Math. Chem</i>. <b>36</b> (4): <span class="nowrap">409–</span>421. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AJOMC.0000044526.22457.bb">10.1023/B:JOMC.0000044526.22457.bb</a>.</cite></li>
<li><cite id="CITEREFLikos2001" class="citation journal cs1">Likos, C. N. (2001). "Effective interactions in soft condensed matter physics". <i>Physics Reports</i>. <b>348</b> (<span class="nowrap">4–</span>5): <span class="nowrap">267–</span>439. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001PhR...348..267L">2001PhR...348..267L</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.473.7668">10.1.1.473.7668</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-1573%2800%2900141-1">10.1016/S0370-1573(00)00141-1</a>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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