Let's define a simple equation $\Omega(x,y)$

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

Then some rules

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I use the rules as

Simplify[Ωx /. rule1 /. rule2]

I get the following output

As you can see, the rules work well in the second term, while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the above-mentioned rules?

I define a simple expression $\Omega$:

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

I also define some rules:

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I try to simplify it after applying my rules; i.e.,

Simplify[Ωx /. rule1 /. rule2]

I get the following output

As you can see, the rules work well in the second term, while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the above-mentioned rules?

Let's define a simple equation $\Omega(x,y)$

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

Then some rules

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I use the rules as

Simplify[Ωx /. rule1 /. rule2]

I get the following

As you can see, the rules work well in the second term, while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the above-mentioned rules?

Let's define a simple equation $\Omega(x,y)$

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

Then some rules

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I use the rules as

Simplify[Ωx /. rule1 /. rule2]

I get the following output

As you can see, the rules work well in the second term, while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the above-mentioned rules?

Let's define a simple equation $\Omega(x,y)$

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

Then some rules

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I use the rules as

Simplify[Ωx /. rule1 /. rule2]

I get the following

As you can see the rules work well in the second term. while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the rules?

Let's define a simple equation $\Omega(x,y)$

x1 = m3*Sqrt[3];
y1 = 0;
x2 = Sqrt[3]/2*(2*m3 - 1);
y2 = 1/2;
x3 = x2;
y3 = -y2;

r1 = Sqrt[(x - x1)^2 + (y - y1)^2];
r2 = Sqrt[(x - x2)^2 + (y - y2)^2];  
r3 = Sqrt[(x - x3)^2 + (y - y3)^2];

Ω = (1 - 2*m3)/r1 + m3/r2 + m3/r3 + 1/2*(x^2 + y^2);

Then some rules

rule1 = {m3*Sqrt[3] -> x10, 1/2*Sqrt[3]*(2*m3 - 1) -> x20};
rule2 = {((x - x10)^2 + y^2) -> r10, ((x - x20)^2 + (y - 1/2)^2) -> r20,
         ((x - x20)^2 + (y + 1/2)^2) -> r30};

When I compute the first derivative with respect to $x$

Ωx = D[Ω, x];

and I use the rules as

Simplify[Ωx /. rule1 /. rule2]

I get the following

As you can see, the rules work well in the second term, while on the other hand they fail in the third and fourth term. Why?

Any suggestions on how to simplify the output according to the above-mentioned rules?