Definition and Properties of Conserved Quantities
– A conserved quantity remains constant over time in a system.
– In mathematics, a conserved quantity is defined as a function of the dependent variables.
– Not all systems have conserved quantities.
– Conserved quantities are not unique and can be produced by applying suitable functions.
– Conserved quantities commonly exist in mathematical models of physical systems.
– In a first-order system of differential equations, a conserved quantity is represented by a scalar-valued function.
– The derivative of the conserved quantity with respect to time is zero.
– The definition of a conserved quantity can be written as the dot product of the gradient of the conserved quantity and the vector function of the system is zero.
– The definition can help in finding conserved quantities or determining their existence.
– The definition is specific to the system and contains system-specific information.
Conserved Quantities in Hamiltonian Mechanics
– In Hamiltonian mechanics, a conserved quantity is defined by a function of the generalized coordinates and generalized momenta.
– The time evolution of the conserved quantity is given by a specific equation.
– The conserved quantity is zero if the Poisson bracket of the function and the Hamiltonian, plus the partial derivative with respect to time, is zero.
– The Poisson bracket represents a mathematical operation.
– Conserved quantities in Hamiltonian mechanics are determined by the properties of the system.
Conserved Quantities in Lagrangian Mechanics
– In Lagrangian mechanics, the energy of a system is conserved if the Lagrangian has no explicit time dependence.
– The energy is defined by a specific equation involving the generalized coordinates and their derivatives.
– If the Lagrangian has no dependence on a specific coordinate, then the corresponding generalized momentum is conserved.
– The conservation of momentum can be derived using the Euler-Lagrange equations.
– The conservation of energy and momentum in Lagrangian mechanics is based on the properties of the system.
Related Concepts
– Conservative system is related to the concept of conserved quantities.
– Lyapunov function is another related concept.
– Hamiltonian system is connected to the study of conserved quantities.
– Conservation law is a broader concept that includes the idea of conserved quantities.
– Noether’s theorem is a fundamental result in physics related to symmetries and conserved quantities.
– Charge and invariant are also related concepts in the context of conserved quantities.
Miscellaneous Points
– Conserved quantities are not unique and can be produced by applying suitable functions.
– Conserved quantities commonly exist in mathematical models of physical systems.
– The definition of a conserved quantity is specific to the system and contains system-specific information. Source: https://en.wikipedia.org/wiki/Conserved_quantity
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A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are not unique, since one can always produce another such quantity by applying a suitable function, such as adding a constant, to a conserved quantity.
Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.
