Question

Texts: The conjugate momenta (pᵣ, pᵩ, and pᵧ) are defined as follows: pᵣ = ṙ = r² pᵩ = ṙ = r² pᵧ = λ̇ cos²(ᵩ) where r is the radial distance and ᵩ and λ are angles. These terms can be used to describe the motion of a satellite under Earth's gravitational potential, according to the following equations: dt/dr = pᵣ = -r²μ + r³pᵩ² + r³cos²(ᵩ)pᵧ² dt/dᵩ = pᵩ = -r²pᵧ²cos³(ᵩ)sin(ᵩ) dt/dλ = pᵧ = r²cos²(ᵩ)pᵧ where μ is a constant and dim[μ] = L³T⁻², and t is the time evolution of the system. Note, you do not need to solve these equations or know how to obtain them to analyze their dimensions. (i) Obtain the dimensions of the three conjugate momenta. (ii) Show that the system of equations that describes the motion of a satellite is dimensionally homogeneous.

          Texts: The conjugate momenta (pᵣ, pᵩ, and pᵧ) are defined as follows:  
pᵣ = ṙ = r²  
pᵩ = ṙ = r²  
pᵧ = λ̇ cos²(ᵩ)  
where r is the radial distance and ᵩ and λ are angles. These terms can be used to describe the motion of a satellite under Earth's gravitational potential, according to the following equations:  
dt/dr = pᵣ = -r²μ + r³pᵩ² + r³cos²(ᵩ)pᵧ²  
dt/dᵩ = pᵩ = -r²pᵧ²cos³(ᵩ)sin(ᵩ)  
dt/dλ = pᵧ = r²cos²(ᵩ)pᵧ  
where μ is a constant and dim[μ] = L³T⁻², and t is the time evolution of the system. Note, you do not need to solve these equations or know how to obtain them to analyze their dimensions. (i) Obtain the dimensions of the three conjugate momenta. (ii) Show that the system of equations that describes the motion of a satellite is dimensionally homogeneous.

Added by Jessica G.

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