Can someone please show the steps for deriving the total energy of a particle from the Polar form of Newton's Second Law?

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Can someone please show the steps for deriving the total energy of a particle from the Polar form of Newton&#039;s Second Law?

Jordan Owen · Accepted Answer

The total energy is   T  +  V, with T kinetic and V potential, and by energy conservation   0  =            T      ˙        +            V      ˙      . Taking T to depend only on x and V only on             x      ˙      ,   0  =            x      ¨        ⋅      ∇                  x        ˙              T  +            x      ˙        ⋅      ∇    x    V. We want this to be equivalent to   0  =  m            x      ¨        +      ∇    x    V, so       ∇                  x        ˙              T  =  m            x      ˙      . The convention that   T  (  0  )  =  0 gives   T  =            m                                    x            ˙                          2              2  This argument doesn&#039;t commit us to Cartesian coordinates; scalar products, including squared lengths, are unchanged when we switch to polar coordinates. In three dimensions,  T  =      m    2        (                            r          ˙                    2        +          r      2                                θ          ˙                    2        +          r      2              sin      2        ⁡    θ                            ϕ          ˙                    2        )    .The last term is deleted in two dimensions, to which we typically switch if radial forces&#039; angular momentum conservation implies the motion is confined to a plane.

Can someone please show the steps for deriving the total energy of a particle from the Polar form of Newton's Second Law?

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