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Least action and partial order

0

$begingroup$

I am fascinated by minimizing principles; my favourite is the least action principle,
$$
S[q]=int_t_1^t_2L(q,dotq) dt ,
$$
which states that the trajectory of a physical system will be such that the action is minimum. The action, $S$, is a function that takes an entire trajectory $gamma: R to M$; $M$ a manifold, and returns a real number.

I understand the minimization problem thusly: $q_cl$ is a minimum of the action, if there exists a neighbourhood of $q_cl$, call in $N_c$ (in the function space of all trajectories) such that for all $qin N_c$ $S[q]geq S[q_cl]$.

The codomain, being a partially ordered set, induces a partial order in the domain; namely, in the set of all trajectories, thus allowing us to label one trajectory as the one that minimizes. Is the structure of partial order ($leq$) in the codomain the weakest possible that allows us to talk about minimization problems?

What I want really to understand, is successively, the contribution of each structure of the codomain to the problem. For example, the partial order of the codomain, allows us to state the problem of minimization. What about the topology of the codomain? What if the topology of the codomain is not metrizable? Completeness, smoothness, etc...?

general-topology optimization manifolds maxima-minima

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asked Mar 19 at 11:19

EEEBEEEB

53139

$endgroup$

add a comment | 

0

$begingroup$

I am fascinated by minimizing principles; my favourite is the least action principle,
$$
S[q]=int_t_1^t_2L(q,dotq) dt ,
$$
which states that the trajectory of a physical system will be such that the action is minimum. The action, $S$, is a function that takes an entire trajectory $gamma: R to M$; $M$ a manifold, and returns a real number.

I understand the minimization problem thusly: $q_cl$ is a minimum of the action, if there exists a neighbourhood of $q_cl$, call in $N_c$ (in the function space of all trajectories) such that for all $qin N_c$ $S[q]geq S[q_cl]$.

The codomain, being a partially ordered set, induces a partial order in the domain; namely, in the set of all trajectories, thus allowing us to label one trajectory as the one that minimizes. Is the structure of partial order ($leq$) in the codomain the weakest possible that allows us to talk about minimization problems?

What I want really to understand, is successively, the contribution of each structure of the codomain to the problem. For example, the partial order of the codomain, allows us to state the problem of minimization. What about the topology of the codomain? What if the topology of the codomain is not metrizable? Completeness, smoothness, etc...?

general-topology optimization manifolds maxima-minima

share|cite|improve this question

asked Mar 19 at 11:19

EEEBEEEB

53139

$endgroup$

add a comment | 

$begingroup$

I am fascinated by minimizing principles; my favourite is the least action principle,
$$
S[q]=int_t_1^t_2L(q,dotq) dt ,
$$
which states that the trajectory of a physical system will be such that the action is minimum. The action, $S$, is a function that takes an entire trajectory $gamma: R to M$; $M$ a manifold, and returns a real number.

I understand the minimization problem thusly: $q_cl$ is a minimum of the action, if there exists a neighbourhood of $q_cl$, call in $N_c$ (in the function space of all trajectories) such that for all $qin N_c$ $S[q]geq S[q_cl]$.

The codomain, being a partially ordered set, induces a partial order in the domain; namely, in the set of all trajectories, thus allowing us to label one trajectory as the one that minimizes. Is the structure of partial order ($leq$) in the codomain the weakest possible that allows us to talk about minimization problems?

What I want really to understand, is successively, the contribution of each structure of the codomain to the problem. For example, the partial order of the codomain, allows us to state the problem of minimization. What about the topology of the codomain? What if the topology of the codomain is not metrizable? Completeness, smoothness, etc...?

general-topology optimization manifolds maxima-minima

share|cite|improve this question

asked Mar 19 at 11:19

EEEBEEEB

53139

$endgroup$

share|cite|improve this question

asked Mar 19 at 11:19

EEEBEEEB

53139

share|cite|improve this question

asked Mar 19 at 11:19

EEEBEEEB

53139

asked Mar 19 at 11:19

EEEBEEEB

53139

asked Mar 19 at 11:19

EEEBEEEB

53139