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Are self-intersecting paths in configuration space allowed by the principle of least action?
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The principle of least action in classical mechanics (with the Lagrangian formalism) states:
Theorem. A path $gamma$ between configurations $q_(1)$ at time $t_1$ and $q_(2)$ at time $t_2$ is a solution to the Euler-Lagrange equations associated to a Lagrangian $mathcal L$ if and only if it is a stationary point of the action functional $$I_mathcal L[gamma] := int_t_1^t_2mathcal L(gamma,gamma',t) dt$$
associated with the Lagrangian $mathcal L$.
When illustrating this principle, people tend to make drawings like the following:
$qquadqquadquad$
All of these synchronous paths do go from configuration $q_(1)$ at time $t_1$ to configuration $q_(2)$ at time $t_1$. However, shoulnd't the self-intersecting ones not be allowed?
If a curve passes through configuration $bar q$ at two different times $t_a$ and $t_b$, it will do so in general at two different generalized velocities $dot q_a$ and $dot q_b$. But the principle does not guarantee the existence (nor the uniqueness) of a curve $gamma$, and instead assumes its existence by hypothesis: to know that the principle may be applied, i.e. that a solution $gamma$ of the Euler-Lagrange equations exists and is unique, we need to assume that the two configurations $q_(1)$ and $q_(2)$ are very (infinitesimally) close in time, so that the usual initial-value existence theorems from the theory of ordinary differential equations may be applied.
This means that, in case the self-intersecting trajectory $gamma$ were to be a solution in this setting, the generalized velocities $dot q_a$ and $dot q_b$ would be achieved within an infinitesimally long span of time, that is, the system would be at configuration $bar q$ and have two different generalized velocities – a very un-physical situation!
Is my reasoning correct?
classical-mechanics
asked Jul 28 at 21:37
giobrach
2,504418
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up vote
2
down vote
favorite
The principle of least action in classical mechanics (with the Lagrangian formalism) states:
Theorem. A path $gamma$ between configurations $q_(1)$ at time $t_1$ and $q_(2)$ at time $t_2$ is a solution to the Euler-Lagrange equations associated to a Lagrangian $mathcal L$ if and only if it is a stationary point of the action functional $$I_mathcal L[gamma] := int_t_1^t_2mathcal L(gamma,gamma',t) dt$$
associated with the Lagrangian $mathcal L$.
When illustrating this principle, people tend to make drawings like the following:
$qquadqquadquad$
All of these synchronous paths do go from configuration $q_(1)$ at time $t_1$ to configuration $q_(2)$ at time $t_1$. However, shoulnd't the self-intersecting ones not be allowed?
If a curve passes through configuration $bar q$ at two different times $t_a$ and $t_b$, it will do so in general at two different generalized velocities $dot q_a$ and $dot q_b$. But the principle does not guarantee the existence (nor the uniqueness) of a curve $gamma$, and instead assumes its existence by hypothesis: to know that the principle may be applied, i.e. that a solution $gamma$ of the Euler-Lagrange equations exists and is unique, we need to assume that the two configurations $q_(1)$ and $q_(2)$ are very (infinitesimally) close in time, so that the usual initial-value existence theorems from the theory of ordinary differential equations may be applied.
This means that, in case the self-intersecting trajectory $gamma$ were to be a solution in this setting, the generalized velocities $dot q_a$ and $dot q_b$ would be achieved within an infinitesimally long span of time, that is, the system would be at configuration $bar q$ and have two different generalized velocities – a very un-physical situation!
Is my reasoning correct?
classical-mechanics
asked Jul 28 at 21:37
giobrach
2,504418
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The principle of least action in classical mechanics (with the Lagrangian formalism) states:
Theorem. A path $gamma$ between configurations $q_(1)$ at time $t_1$ and $q_(2)$ at time $t_2$ is a solution to the Euler-Lagrange equations associated to a Lagrangian $mathcal L$ if and only if it is a stationary point of the action functional $$I_mathcal L[gamma] := int_t_1^t_2mathcal L(gamma,gamma',t) dt$$
associated with the Lagrangian $mathcal L$.
When illustrating this principle, people tend to make drawings like the following:
$qquadqquadquad$
All of these synchronous paths do go from configuration $q_(1)$ at time $t_1$ to configuration $q_(2)$ at time $t_1$. However, shoulnd't the self-intersecting ones not be allowed?
If a curve passes through configuration $bar q$ at two different times $t_a$ and $t_b$, it will do so in general at two different generalized velocities $dot q_a$ and $dot q_b$. But the principle does not guarantee the existence (nor the uniqueness) of a curve $gamma$, and instead assumes its existence by hypothesis: to know that the principle may be applied, i.e. that a solution $gamma$ of the Euler-Lagrange equations exists and is unique, we need to assume that the two configurations $q_(1)$ and $q_(2)$ are very (infinitesimally) close in time, so that the usual initial-value existence theorems from the theory of ordinary differential equations may be applied.
This means that, in case the self-intersecting trajectory $gamma$ were to be a solution in this setting, the generalized velocities $dot q_a$ and $dot q_b$ would be achieved within an infinitesimally long span of time, that is, the system would be at configuration $bar q$ and have two different generalized velocities – a very un-physical situation!
Is my reasoning correct?
classical-mechanics
asked Jul 28 at 21:37
giobrach
2,504418
The principle of least action in classical mechanics (with the Lagrangian formalism) states:
Theorem. A path $gamma$ between configurations $q_(1)$ at time $t_1$ and $q_(2)$ at time $t_2$ is a solution to the Euler-Lagrange equations associated to a Lagrangian $mathcal L$ if and only if it is a stationary point of the action functional $$I_mathcal L[gamma] := int_t_1^t_2mathcal L(gamma,gamma',t) dt$$
associated with the Lagrangian $mathcal L$.
When illustrating this principle, people tend to make drawings like the following:
$qquadqquadquad$
All of these synchronous paths do go from configuration $q_(1)$ at time $t_1$ to configuration $q_(2)$ at time $t_1$. However, shoulnd't the self-intersecting ones not be allowed?
If a curve passes through configuration $bar q$ at two different times $t_a$ and $t_b$, it will do so in general at two different generalized velocities $dot q_a$ and $dot q_b$. But the principle does not guarantee the existence (nor the uniqueness) of a curve $gamma$, and instead assumes its existence by hypothesis: to know that the principle may be applied, i.e. that a solution $gamma$ of the Euler-Lagrange equations exists and is unique, we need to assume that the two configurations $q_(1)$ and $q_(2)$ are very (infinitesimally) close in time, so that the usual initial-value existence theorems from the theory of ordinary differential equations may be applied.
This means that, in case the self-intersecting trajectory $gamma$ were to be a solution in this setting, the generalized velocities $dot q_a$ and $dot q_b$ would be achieved within an infinitesimally long span of time, that is, the system would be at configuration $bar q$ and have two different generalized velocities – a very un-physical situation!
Is my reasoning correct?
classical-mechanics
asked Jul 28 at 21:37
giobrach
2,504418
asked Jul 28 at 21:37
giobrach
2,504418
asked Jul 28 at 21:37
giobrach
2,504418
asked Jul 28 at 21:37
giobrach
2,504418
2,504418
add a comment |Â
add a comment |Â
1 Answer
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Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $mathbbS^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.
answered Jul 29 at 14:43
Qmechanic
4,39411746
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $mathbbS^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.
answered Jul 29 at 14:43
Qmechanic
4,39411746
add a comment |Â
up vote
0
down vote
Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $mathbbS^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.
answered Jul 29 at 14:43
Qmechanic
4,39411746
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $mathbbS^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.
answered Jul 29 at 14:43
Qmechanic
4,39411746
Yes, self-intersections are allowed, both for stationary and non-stationary paths. E.g. if the configuration space is topological non-trivial. Consider e.g. a circle $mathbbS^1$. Then the EL equation with pertinent Dirichlet boundary conditions (BC) could have infinitely many solutions (often called instantons in physics jargon) because of different winding sectors.
answered Jul 29 at 14:43
Qmechanic
4,39411746
answered Jul 29 at 14:43
Qmechanic
4,39411746
answered Jul 29 at 14:43
Qmechanic
4,39411746
answered Jul 29 at 14:43
Qmechanic
4,39411746
4,39411746
add a comment |Â
add a comment |Â
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