Question

Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Specifically, why can't we use tug, the third derivative of momentum, to solve physics problems even though it has a conservation principle? Is it too hard to find tugs in the physical world?

Derivations

Let's define,

$$(1) \quad P_n=\cfrac{d^n p}{dt^n}$$

Where $p$ is momentum. If we take the integral of $(1)$ over a path $C$ parameterized by $r(t)$, and call the result $W_n$, we get,

$$(2) \quad W_n=\int_C P_n(r(t)) \ dr$$

$(2)$ can be simplified a bit to the form,

$$(3) \quad W_n=\int_{t_i}^{t_f} P_n(r(t)) \cdot \dot r(t) \ dt$$ $$\Rightarrow W_n=\int_{t_i}^{t_f} m \cdot \cfrac{d^{n} (\dot r(t))}{dt^{n}} \cdot \dot r(t) \ dt$$

For $n=0,2,4...$ the expressions for $W_n$ simplify to,

$$W_0=\Delta K_0=\int_{t_i}^{t_f} m \cdot (\cdot r(t))^2 \ dt$$ $$W_2=\Delta K_2=\cfrac{1}{2} \cdot m \cdot \dot r(t_f)^2-\cfrac{1}{2} \cdot m \cdot \dot r(t_i)^2$$ $$W_4=\Delta K_4=m \cdot \dot r(t) \cdot \cfrac{d^{2} (\dot r(t))}{dt^{2}}-\cfrac{1}{2} \cdot m \cdot \cfrac{d (\dot r(t))}{dt}$$

The analogy to Kinetic energy should be apparent. In addition, I'd expect an analogy for Potential energy to exist. If we assume these functions exist then,

$$(4) \quad W_n=\Delta K_n=-\Delta P_n$$

Which immediately leads to,

$$\Delta K_n+\Delta P_n=0$$

And then finally,

$$\lambda=K_i+P_i=K_f+P_f$$

However, I am at a loss as how one would actually find real life $P_n$ where $n \gt 1$. For instance, how would deriving the tug of the gravitational field work?

Question

Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Even if they aren't needed, could it be done in theory? For instance, why can't we use tug, the third derivative of momentum, to solve physics problems even though it has a conservation principle? Is it too hard to find tugs in the physical world?

Derivations

Let's define,

$$(1) \quad P_n=\cfrac{d^n p}{dt^n}$$

Where $p$ is momentum. If we take the integral of $(1)$ over a path $C$ parameterized by $r(t)$, and call the result $W_n$, we get,

$$(2) \quad W_n=\int_C P_n(r(t)) \ dr$$

$(2)$ can be simplified a bit to the form,

$$(3) \quad W_n=\int_{t_i}^{t_f} P_n(r(t)) \cdot \dot r(t) \ dt$$ $$\Rightarrow W_n=\int_{t_i}^{t_f} m \cdot \cfrac{d^{n} (\dot r(t))}{dt^{n}} \cdot \dot r(t) \ dt$$

For $n=0,2,4...$ the expressions for $W_n$ simplify to,

$$W_0=\Delta K_0=\int_{t_i}^{t_f} m \cdot (\cdot r(t))^2 \ dt$$ $$W_2=\Delta K_2=\cfrac{1}{2} \cdot m \cdot \dot r(t_f)^2-\cfrac{1}{2} \cdot m \cdot \dot r(t_i)^2$$ $$W_4=\Delta K_4=m \cdot \dot r(t) \cdot \cfrac{d^{2} (\dot r(t))}{dt^{2}}-\cfrac{1}{2} \cdot m \cdot \cfrac{d (\dot r(t))}{dt}$$

The analogy to Kinetic energy should be apparent. In addition, I'd expect an analogy for Potential energy to exist. If we assume these functions exist then,

$$(4) \quad W_n=\Delta K_n=-\Delta P_n$$

Which immediately leads to,

$$\Delta K_n+\Delta P_n=0$$

And then finally,

$$\lambda=K_i+P_i=K_f+P_f$$

However, I am at a loss as how one would actually find real life $P_n$ where $n \gt 1$. For instance, how would deriving the tug of the gravitational field work?