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Dissipative Force

Dissipative Force. Generalized force came from a transformation. Jacobian transformation Not a constraint Conservative forces were separated in the Lagrangian. Non-Potential Force. Velocity Dependent. A function M may exist that still permits a Lagrangian.

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Dissipative Force

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  1. Dissipative Force

  2. Generalized force came from a transformation. Jacobian transformation Not a constraint Conservative forces were separated in the Lagrangian. Non-Potential Force

  3. Velocity Dependent • A function M may exist that still permits a Lagrangian. • Requires force to remain after Lagrange equation • M is a generalized potential • The Lagrangian must include this to permit solutions by the usual equation.

  4. The electromagnetic force depends on velocity. Manifold is TQ Both E and B derive from potentials f, A. Generalized potential M Use this in a Lagrangian, test to see that it returns usual result. Electromagnetic Potential

  5. Electromagnetic Lagrangian matching Newtonian equation

  6. Dissipative forces can’t be treated with a generalized potential. Potential forces in L Non-potential forces to the right Friction is a non-potential force. Linear in velocity Could be derived from a velocity potential Dissipative Force

  7. Dissipative forces can be treated if they are linear in velocity. This is the Rayleigh dissipation function. Lagrange’s equations then include dissipative force. Rayleigh Function

  8. Energy Lost • The Rayleigh function is related to the energy lost. • Work done is related to power • Power is twice the Rayleigh function

  9. Example The 1-D damped harmonic oscillator has linear velocity dependence. Rayleigh function from damping The power lost from Rayleigh Use damped oscillator solution to compare with time. Damped Oscillator next

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