| Summary: | We consider the two planes at zero temperature with isotropic conductivity that are in relative lateral motion with velocity <i>v</i> and interplane distance <i>a</i>. Two models of conductivity are taken into account—the constant and frequency-dependent Drude models. The normal (perpendicular to planes) Casimir force is analyzed in detail for two systems—(i) two planes with identical conductivity and (ii) one plane that is a perfect metal. The velocity correction to the Casimir energy, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>v</mi></msub><mi mathvariant="script">E</mi></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∝</mo><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>v</mi><mn>2</mn></msup></semantics></math></inline-formula>, for small enough velocities is used for all considered cases. In the case of constant conductivity, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>, the energy correction is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>v</mi></msub><mi mathvariant="script">E</mi><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∝</mo><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>/</mo><msup><mi>a</mi><mn>3</mn></msup><msup><mfenced separators="" open="(" close=")"><mi>v</mi><mo>/</mo><mi>η</mi></mfenced><mn>2</mn></msup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>≪</mo><mi>η</mi><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula>.
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