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I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate

[tex] \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0 [/tex]

[tex] \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{1}{\rho}\frac{\partial p}{\partial x} = 0 [/tex]

need to be converted to,

[tex] \frac{\partial v}{\partial t} - \frac{\partial u}{\partial h} = 0 [/tex]

[tex] \frac{\partial u}{\partial t} + \frac{\partial p}{\partial h} = 0 [/tex]

where the Lagrangian mass coordinate has the relation,

[tex] \frac{\partial h}{\partial x} = \rho [/tex]

and

[tex] v = \frac{1}{\rho} [/tex]

Thanks.

*h*. The one dimensional Euler equations given by,[tex] \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0 [/tex]

[tex] \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{1}{\rho}\frac{\partial p}{\partial x} = 0 [/tex]

need to be converted to,

[tex] \frac{\partial v}{\partial t} - \frac{\partial u}{\partial h} = 0 [/tex]

[tex] \frac{\partial u}{\partial t} + \frac{\partial p}{\partial h} = 0 [/tex]

where the Lagrangian mass coordinate has the relation,

[tex] \frac{\partial h}{\partial x} = \rho [/tex]

and

[tex] v = \frac{1}{\rho} [/tex]

Thanks.

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