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The transition between the Hamiltonian and Lagrangian formalisms in mechanics can be accomplished by means of the Hamilton-Jacobi theory. Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t_0, t_1)$ satisfying the Hamilton-Jacobi equation:

$\displaystyle{\frac{\partial S}{\partial t}+H\left(x_1,\frac{\partial S}{\partial x_1}, t\right) = 0}$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initial and final coordinates as:

$\displaystyle{F_t(x_0, x_1) = f\left(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) \right)}$

The trasnsformation Jacobian is given by:

$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $

And the normalization condition:

$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable

The transition between the Hamiltonian and Lagrangian formalisms in Mechanics can be accomplished by means of the Hamilton-Jacobi theory. 

Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t_0, t_1)$ satisfying the Hamilton-Jacobi equation:

$$\displaystyle{\frac{\partial S}{\partial t}+H\left(x_1,\frac{\partial S}{\partial x_1}, t\right) = 0}$$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initial and final coordinates as:

$$\displaystyle{F_t(x_0, x_1) = f\left(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) \right)}$$

The transformation Jacobian is given by:

$$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $$

And the normalization condition:

$$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable

The transition between the Hamiltonian and Lagrangian formalisms in mechanics can be accomplished by means of the Hamilton-Jacobi theory. Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t)$ satisfying the hamilton Jacobi equation:

$\displaystyle{\frac{\partial S}{\partial t}+H\left(x,\frac{\partial S}{\partial x}, t\right) = 0}$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initial and final coordinates as:

$\displaystyle{F_t(x_0, x_1) = f\left(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) \right)}$

The trasnsformation Jacobian is given by:

$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $

And the normalization condition:

$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable

The transition between the Hamiltonian and Lagrangian formalisms in mechanics can be accomplished by means of the Hamilton-Jacobi theory. Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t_0, t_1)$ satisfying the Hamilton-Jacobi equation:

$\displaystyle{\frac{\partial S}{\partial t}+H\left(x_1,\frac{\partial S}{\partial x_1}, t\right) = 0}$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initial and final coordinates as:

$\displaystyle{F_t(x_0, x_1) = f\left(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) \right)}$

The trasnsformation Jacobian is given by:

$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $

And the normalization condition:

$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable

The transition between the Hamiltonian and Lagrangian formalisms in mechanics can be accomplished by means of the Hamilton-Jacobi theory. Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t)$ satisfying the hamilton Jacobi equation:

$\displaystyle{\frac{\partial S}{\partial t}+H(x,\frac{\partial S}{\partial x}, t) = 0}$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initail and final coordinates as:

$\displaystyle{F_t(x_0, x_1) = f(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) )}$

The trasnsformation Jacobian is given by:

$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $

And the normalization condition:

$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable

The transition between the Hamiltonian and Lagrangian formalisms in mechanics can be accomplished by means of the Hamilton-Jacobi theory. Consider for example a classical statistical ensemble on a phase space $(x,p)$ defined by:

A. The (initial) state of this ensemble is defined by a distribution function $f(x_0,p_0)$ satisfying the normalization condition:

$\displaystyle{\int f(x_0,p_0) dx_0dp_0 = 1}$

($(x_0,p_0)$ are the initial conditions)

B. The time evolution is governed by the Hamiltonian function $H(x,p, t)$.

According to the Hamilton-Jacobi theory, there exists Hamilton-Jacobi phase function $S(x_0, x_1, t)$ satisfying the hamilton Jacobi equation:

$\displaystyle{\frac{\partial S}{\partial t}+H\left(x,\frac{\partial S}{\partial x}, t\right) = 0}$

(where $(x_1,p_1)$ are the coordinates and momenta at time $t$)

The momenta can be derived from the Hamilton-Jacobi phase function:

$\displaystyle{p_i = \frac{\partial S}{\partial x_i}}$

The problem of expressing the state of the system in terms of the initial and final coordinates is rendered to a problem of transformation of probability distributions. We can define the state of the system in the initial and final coordinates as:

$\displaystyle{F_t(x_0, x_1) = f\left(x_0,\frac{\partial S}{\partial x_1}(x_0, x_1, t) \right)}$

The trasnsformation Jacobian is given by:

$ \displaystyle{dx_0 dp_0 = \frac{\partial^2 S}{\partial x_0\partial x_1}}dx_0 dx_1 $

And the normalization condition:

$\displaystyle{\int F_t(x_0, x_1) \frac{\partial^2 S}{\partial x_0\partial x_1}(x_0, x_1, t)dx_0dx_1 = 1}$

In the general case, the joint distribution $F_t(x_0, x_1)$ will not be separable