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If the Swedish invasion of Russia would have happened 1700-1701, the circumstances for the Swedes wouldn't have been worse than they were in 1708-1709 at least. Charles was already gambling, when he decided to attack the Commonwealth and left his rear open for the Russians. So, why not to do it vice versa, it wouldn't have included any more risk to leave the rear open for Commonwealth and secure the supply with the garrisons of Riga and Narva and raiding the depo in St.Petersburgh constructions. For real Charles invaded Russia in winter 1708-1709 and it was a terrible winter, in 1700-1701 the winter was mild. Besides, Charles wouldn't have faced opposing forces, because the enemy was defeated and could not have practiced scorched earth -tactics.

In summer/autumn 1701 Charles would have found his allies from the Ottomans and the Cossacks of Ukraine. This alliance Sweden-the Ottomans-Cossacks of Ukraine could have defeated the remaining enemies, the Commonwealth and Denmark.

Here is the map
https://upload.wikimedia.org/wikipedia/commons/0/08/Great_Northern_War_Part1.png

his rear was open, but the Russians were beaten and the Commonwealth was besieging Riga (the second largest city he had). Had they succeed they could storm the Baltic Provinces which is a game over for Sweden. Therefore he have to move there and relieve it. Until he neutralized the Polish-Saxon army which is about Grodno and Fraustadt he was always under the threat of getting cut from his bases in Sweden.
 
Here is the map
https://upload.wikimedia.org/wikipedia/commons/0/08/Great_Northern_War_Part1.png

his rear was open, but the Russians were beaten and the Commonwealth was besieging Riga (the second largest city he had). Had they succeed they could storm the Baltic Provinces which is a game over for Sweden. Therefore he have to move there and relieve it. Until he neutralized the Polish-Saxon army which is about Grodno and Fraustadt he was always under the threat of getting cut from his bases in Sweden.

I agree, that Riga had to be relieved. But after that, there was no necessary need for the Commonwealth campaign, Charles could have chose the Russian invasion now as well, and that's what his military advisors told him to do. But Charles kept his head and we all know what happened.
 
A small correction. The Commonwealth wasn't at war with Sweden at all. Augustus II attacked Sweden in his capacity as Prince-Elector of Saxony, while the Commonwealth's Sejm politely told him that the war was against Polish-Lithuanian interests. In fact, Polish-Lithuanian magnates wanted to go against either Russia or Brandenburg. The former choice would make Commonwealth an ally of Sweden.

How did Augustus II manage to attack Sweden by land without Commonwealth cooperation? Well, a brief civil war in Lithuania erupted between two noble faction. It was over before anyone even got wind that it started, but Augustus nobly and selflessly volunteered to send in some soldiers to maintain peace. But because the officers in the Commonwealth's army often were related or associated to one party or another, the Sejm agreed that impartial Saxon troops would be sent in. And that's how Augustus managed to have an army ready to attack Sweden.

Things changed only after Charles XII invaded the Commonwealth. By then Augustus was so unpopular that much of the army fought half-hearted and scores of nobles abandoned the king altogether.
 
A small correction. The Commonwealth wasn't at war with Sweden at all. Augustus II attacked Sweden in his capacity as Prince-Elector of Saxony, while the Commonwealth's Sejm politely told him that the war was against Polish-Lithuanian interests. In fact, Polish-Lithuanian magnates wanted to go against either Russia or Brandenburg. The former choice would make Commonwealth an ally of Sweden.

How did Augustus II manage to attack Sweden by land without Commonwealth cooperation? Well, a brief civil war in Lithuania erupted between two noble faction. It was over before anyone even got wind that it started, but Augustus nobly and selflessly volunteered to send in some soldiers to maintain peace. But because the officers in the Commonwealth's army often were related or associated to one party or another, the Sejm agreed that impartial Saxon troops would be sent in. And that's how Augustus managed to have an army ready to attack Sweden.

Things changed only after Charles XII invaded the Commonwealth. By then Augustus was so unpopular that much of the army fought half-hearted and scores of nobles abandoned the king altogether.

Yes, my apologies for everyone, I have generalized falsely in the issues considering this thread and I should have mentioned Electorate of Saxony instead the Commonwealth in some cases. This is somehow complicated, because the Polish-Lithuanian Commonwealth is mentioned as a belligerent of both sides in the history statements of The Great Northern War. But, one more time, sorry.