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We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ it happens. The equations of motions for the dynamics of a point particle are (any relativistic form) of the Newton's laws, which define a second order degree Cauchy problem: as such, given the initial and boundary conditions, velocities can be calculated taking derivatives with respect to the path length $s$ as $v^{\mu}=dx^{\mu}/ds$ and therefore they are not independent variables.

In Hamiltonian and Lagrangian mechanics velocities can be treated as independent degrees of freedom unrelated to the positions, in principle. In fact the description of those systems is realised as a path on a $2n$-dimensional manifold, $n$ being the number of "position-like" degrees of freedom of the particle, which exactly fulfills the question you asked. Such manifold is the tangent (or cotangent) bundle $TM$ of the configuration space, which often undergoes the name of phase space. In general field theories, furthermore, one always defines the variation principle as an action whose variables are the fields $\phi(x)$ and its derivative $\dot{\phi}(x)$ treated as independent variables, giving rise to a twice dimensional dynamical manifold. Quantum field theory and string theory follow in some extends that flavoursflavour too.

We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ it happens. The equations of motions for the dynamics of a point particle are (any relativistic form) of the Newton's laws, which define a second order degree Cauchy problem: as such, given the initial and boundary conditions, velocities can be calculated taking derivatives with respect to the path length $s$ as $v^{\mu}=dx^{\mu}/ds$ and therefore they are not independent variables.

In Hamiltonian and Lagrangian mechanics velocities can be treated as independent degrees of freedom unrelated to the positions, in principle. In fact the description of those systems is realised as a path on a $2n$-dimensional manifold, $n$ being the number of "position-like" degrees of freedom of the particle, which exactly fulfills the question you asked. Such manifold is the tangent (or cotangent) bundle $TM$ of the configuration space, which often undergoes the name of phase space. In general field theories, furthermore, one always defines the variation principle as an action whose variables are the fields $\phi(x)$ and its derivative $\dot{\phi}(x)$ treated as independent variables, giving rise to a twice dimensional dynamical manifold. Quantum field theory and string theory follow in some extends that flavours too.

We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ it happens. The equations of motions for the dynamics of a point particle are (any relativistic form) of the Newton's laws, which define a second order degree Cauchy problem: as such, given the initial and boundary conditions, velocities can be calculated taking derivatives with respect to the path length $s$ as $v^{\mu}=dx^{\mu}/ds$ and therefore they are not independent variables.

In Hamiltonian and Lagrangian mechanics velocities can be treated as independent degrees of freedom unrelated to the positions, in principle. In fact the description of those systems is realised as a path on a $2n$-dimensional manifold, $n$ being the number of "position-like" degrees of freedom of the particle, which exactly fulfills the question you asked. Such manifold is the tangent (or cotangent) bundle $TM$ of the configuration space, which often undergoes the name of phase space. In general field theories, furthermore, one always defines the variation principle as an action whose variables are the fields $\phi(x)$ and its derivative $\dot{\phi}(x)$ treated as independent variables, giving rise to a twice dimensional dynamical manifold. Quantum field theory and string theory follow in some extends that flavour too.

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We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ it happens. The equations of motions for the dynamics of a point particle are (any relativistic form) of the Newton's laws, which define a second order degree Cauchy problem: as such, given the initial and boundary conditions, velocities can be calculated taking derivatives with respect to the path length $s$ as $v^{\mu}=dx^{\mu}/ds$ and therefore they are not independent variables.

In Hamiltonian and Lagrangian mechanics velocities can be treated as independent degrees of freedom unrelated to the positions, in principle. In fact the description of those systems is realised as a path on a $2n$-dimensional manifold, $n$ being the number of "position-like" degrees of freedom of the particle, which exactly fulfills the question you asked. Such manifold is the tangent (or cotangent) bundle $TM$ of the configuration space, which often undergoes the name of phase space. In general field theories, furthermore, one always defines the variation principle as an action whose variables are the fields $\phi(x)$ and its derivative $\dot{\phi}(x)$ treated as independent variables, giving rise to a twice dimensional dynamical manifold. Quantum field theory and string theory follow in some extends that flavours too.